Table Of ContentTotal rainbow connection of digraphs
Hui Lei1, Henry Liu2∗, Colton Magnant3, Yongtang Shi1
7
1 1Center for Combinatorics and LPMC
0
Nankai University, Tianjin 300071, China
2
[email protected], [email protected]
n
a
J 2School of Mathematics and Statistics
6 Central South University, Changsha 410083, China
1
[email protected]
]
O
3Department of Mathematical Sciences
C
Georgia Southern University, Statesboro, GA 30460-8093, USA
.
h [email protected]
t
a
m 29 November 2016
[
1
v
3 Abstract
8
2 An edge-coloured path is rainbow if its edges have distinct colours. For a connected
4 graph G, the rainbow connection number (resp. strong rainbow connection number) of
0 G is the minimum number of colours required to colour the edges of G so that, any
.
1 two vertices of G are connected by a rainbow path (resp. rainbow geodesic). These two
0 graph parameters were introduced by Chartrand, Johns, McKeon and Zhang in 2008.
7 Krivelevichand Yuster generalisedthis concept to the vertex-colouredsetting. Similarly,
1
Liu, Mestre and Sousa introduced the version which involves total-colourings.
:
v Dorbec, Schiermeyer, Sidorowicz and Sopena extended the concept of the rainbow
i connection to digraphs. In this paper, we consider the (strong) total rainbow connection
X
number of digraphs. Results on the (strong) total rainbow connection number of biori-
r
a entations of graphs, tournaments and cactus digraphs are presented.
Keywords: Total rainbow connection; digraphs; tournaments; cactus digraphs; biori-
entation
1 Introduction
All graphs and digraphs considered in this paper are finite and simple. That is, we do not
permit the existence of loops, multiple edges (for graphs), and multiple directed arcs (for
digraphs). We follow the terminology and notation of Bollob´as [4] for those not defined here.
∗Corresponding author
1
The concept of rainbow connection in graphs was introduced by Chartrand et al. [6].
An edge-coloured path is rainbow if its edges have distinct colours. An edge-colouring of a
connected graph G is rainbow connected if any two vertices of G are connected by a rainbow
path. The rainbow connection number of G, denoted by rc(G), is the minimum number
of colours in a rainbow connected edge-colouring of G. An edge-colouring of G is strongly
rainbow connected if for every pair of vertices u and v, there exists a rainbow u−v geodesic,
i.e. a path of length equal to the distance between u and v. The minimum number of colours
in a strongly rainbow connected edge-colouring of G is the strong rainbow connection number
ofG,denotedbysrc(G). Asanaturalcounterparttotherainbowconnectionofedge-coloured
graphs, Krivelevich and Yuster [10] proposed the concept of rainbow vertex-connection. A
vertex-coloured path is vertex-rainbow if its internal vertices have distinct colours. A vertex-
colouring of a connected graph G is rainbow vertex-connected if any two vertices of G are
connected by a vertex-rainbow path. The rainbow vertex-connection number of G, denoted
by rvc(G), is the minimumnumberof colours in a rainbow vertex-connected vertex-colouring
of G. Corresponding to the strong rainbow connection, Li et al. [12] introduced the notion
of strong rainbow vertex-connection. A vertex-colouring of G is strongly rainbow vertex-
connected if for every pair of vertices u and v, there exists a vertex-rainbow u−v geodesic.
The strong rainbow vertex-connection number of G, denoted by srvc(G), is the minimum
number of colours in a strongly rainbow vertex-connected vertex-colouring of G. We refer
the reader to the survey [13] and the monograph [14] on the subject of rainbow connection
in graphs.
Liu et al. [15] proposed the concept of total rainbow connection. A total-coloured path is
total-rainbow if its edges and internal vertices have distinct colours. A total-colouring of a
connected graph G is total rainbow connected if any two vertices are connected by a total-
rainbowpath. Thetotal rainbow connection number ofG, denotedbytrc(G), istheminimum
numberof colours ina total rainbow connected total-colouring of G. Atotal-colouring of Gis
strongly total rainbow connected if any two vertices u and v are connected by a total-rainbow
u−v geodesic. The strong total rainbow connection number of G, denoted by strc(G), is the
minimum number of colours in a strongly total rainbow connected total-colouring of G.
In [8], Dorbec et al. introduced the concept of rainbow connection of digraphs. A directed
path, or simply a path P, is a digraph consisting of a sequence of vertices v0,v1,...,vℓ and
arcs vi−1vi for 1≤ i ≤ ℓ. We also say that P is a v0−vℓ path, and its length is the number of
arcs ℓ. A digraph D is strongly connected if for any order pair of vertices (u,v) in D, there
exists au−v path. An arc-coloured pathis rainbow ifits arcs have distinctcolours. Let D be
a strongly connected digraph. An arc-colouring of D is rainbow connected if for any ordered
pair of vertices (u,v) in D, there is a rainbow u−v path. The rainbow connection number
→
of D, denoted by rc(D), is the minimum number of colours in a rainbow connected arc-
colouring of D. Subsequently, there have been some results on this topic, which considered
many different classes of digraphs [1, 2, 3, 9]. Very recently, Lei et al. [11] introduced the
rainbow vertex-connection of digraphs. A vertex-coloured directed path is vertex-rainbow if
its internalvertices have distinctcolours. Avertex-colouring ofD israinbow vertex-connected
if for any ordered pair of vertices (u,v) in D, there exists a vertex-rainbow u−v path. The
→
rainbow vertex-connection number of D, denoted by rvc(D), is the minimum number of
colours in a rainbow vertex-connected vertex-colouring of D.
2
In this paper, we will study the total rainbow connection number of digraphs. Let D be
a strongly connected digraph. A total-coloured directed path is total-rainbow if its arcs and
internal vertices have distinct colours. A total-colouring of D is total rainbow connected if
for any ordered pair of vertices (u,v) in D, there exists a total-rainbow u − v path. The
→
total rainbow connection number of a digraph D, denoted by trc(D), is the minimum number
of colours in a total rainbow connected total-colouring of D. Likewise, a total-colouring of
D is strongly total rainbow connected if for any ordered pair of vertices (u,v), there exists
a total-rainbow u−v geodesic. The strong total rainbow connection number of D, denoted
→
by strc(D), is the minimum number of colours in a strongly total rainbow connected total-
colouring of D.
This paper will be organised as follows. In Section 2, we present many general results
→ →
about the functions trc(D) and strc(D), as well as their relationships to the functions
→ → → → →
rc(D),src(D),rvc(D) and srvc(D). In Section 3, we consider the functions trc(D) and
→ →
strc(D) for some specific digraphs D. In Section 4, we consider the functions trc(T) and
→ →
strc(T) for tournaments T. Finally in Section 5, we consider the functions rvc(Q) and
→
trc(Q) for cactus digraphs Q.
2 Definitions, remarks, and results for general digraphs
We begin with some definitions about digraphs. For a digraph D, its vertex and arc sets are
denoted by V(D) and A(D). For an arc uv ∈ A(D), we say that v is an out-neighbour of u,
and u is an in-neighbour of v. Moreover, we call uv an in-arc of v and an out-arc of u. We
denote the set of out-neighbours (resp. in-neighbours) of u in D by Γ+(u) (resp. Γ−(u)). Let
Γ[u] = Γ+(u)∪Γ−(u)∪{u}. For u,v ∈ V(D), the distance from u to v (i.e. the length of a
shortest u−v path) in D is denoted by d(u,v), or d (u,v) if we wish to emphasize that the
D
distance is taken in the digraph D. Let diam(D) denote the diameter of D.
If uv,vu ∈ A(D), then we say that uv and vu are symmetric arcs. If uv ∈ A(D) and
vu ∈/ A(D), then uv is an asymmetric arc. The digraph D is an oriented graph if every arc
of D is asymmetric. A tournament is an oriented graph where every two vertices have one
asymmetric arc joining them. A cactus is a strongly connected oriented graph where each
↔
arc belongs to exactly one directed cycle. Given a graph G, its biorientation is the digraphG
→
obtained by replacing each edge uv of G by the pair of symmetric arcs uv and vu. LetP and
n
→ →
C denote the directed path and directed cycle of order n, respectively (where n ≥ 3 forC ),
n n
→ → →
i.e. we may let V(Pn) = V(Cn)= {v0,...,vn−1}, and A(Pn) = {v0v1,v1v2...,vn−2vn−1} and
→ →
A(Cn) = A(Pn)∪ vn−1v0. If C is a directed cycle and u,v ∈ V(C), we write uCv for the
unique u−v directed path in C.
For a subset X ⊂ V(D), we denote by D[X] the subdigraph of D induced by X. Given
two digraphs D and H, and u ∈ V(D), we define Du→H to be the digraph obtained from D
and H by replacing the vertex u by a copy of H, and replacing the arc xu (resp. ux) in D by
all the arcs xv (resp. vx) for v ∈ V(H). We say that Du→H is obtained from D by expanding
u into H. Note that the digraph obtained from D by expanding every vertex into H is also
known as the lexicographic product D◦H.
Now, weshallpresentsomeremarksandbasicresults forthetotal rainbowconnection and
3
strong total rainbow connection numbers, for general digraphs and biorientations of graphs.
We first note that in a total rainbow connected colouring of a strongly connected digraph D,
there must be a path between some two vertices with at least 2diam(D)−1 colours. Thus,
we have the following proposition.
Proposition 1. Let D be a strongly connected digraph with n vertices and m arcs. Then
→ →
2diam(D)−1 ≤ trc(D) ≤ strc(D) ≤ n+m. (1)
↔
It is easy to see that the bioriented paths P , for n ≥ 2, form an infinite family of graphs
n
where we have equalities in the firsttwo inequalities in (1). Also, it is not difficult to see that
→
for every directed cycle C with n ≥ 5, we have equalities in the last two inequalities in (1).
n
These two results will be included in Theorems 11 and 12.
In the next result, we give equivalences and implications between several conditions, when
the rainbow connection parameters are small.
Theorem 2. Let D be a non-trivial, strongly connected digraph.
(a) The following are equivalent.
(i) D is a bioriented complete graph.
(ii) diam(D) = 1.
→
(iii) rc(D) = 1.
→
(iv) src(D) = 1.
→
(v) rvc(D)= 0.
→
(vi) srvc(D) = 0.
→
(vii) trc(D) = 1.
→
(viii) strc(D)= 1.
→ →
(b) strc(D)≥ trc(D) ≥ 3 if and only if D is not a bioriented complete graph.
→ →
(c) (i) rc(D) = 2 if and only if src(D) = 2.
→ →
(ii) rvc(D)= 1, if and only if srvc(D)= 1, if and only if diam(D) = 2.
→ →
(iii) trc(D) = 3 if and only if strc(D)= 3.
Moreover, any of the conditions in (i) implies any of the conditions in (iii), and any of
the conditions in (iii) implies any of the conditions in (ii).
Proof. (a) In [1, 11], it was proved that conditions (i) to (vi) are equivalent. Now clearly, we
have (i) ⇒ (viii), and by (1), it is easy to see that (viii) ⇒ (vii) ⇒ (ii).
→ →
(b) If D is not a bioriented complete graph, then diam(D) ≥ 2, and strc(D) ≥ trc(D) ≥ 3
follows from (1). The converse clearly holds by (a).
(c) Parts (i) and (ii) were proved in [1] and [11] respectively. We first prove (iii). If
→
strc(D) = 3, then by (a), we have D is not a bioriented complete graph. By (b), we have
4
→ → → →
3 ≤ trc(D) ≤ strc(D) = 3, and hence trc(D) = 3. Conversely, suppose that trc(D) = 3.
→
Then (1) implies strc(D) ≥ 3. Also, there exists a total rainbow connected colouring for D,
→
using trc(D) = 3 colours. In such a total-colouring, for any u,v ∈ V(D), either uv ∈ A(D),
oruv 6∈ A(D)andthereisatotal-rainbow u−v pathoflength2, whichisalsoatotal-rainbow
→ →
u−v geodesic. Thus strc(D) ≤ 3, and strc(D) = 3 as required.
Now, we consider the final part of (c). Firstly, suppose that either condition in (i) holds,
→ →
so that rc(D) = 2. Then (a) implies diam(D) ≥ 2, and (1) implies trc(D) ≥ 3. Moreover,
→
there exists a rainbow connected arc-colouring for D, using rc(D) = 2 colours. Clearly by
colouring all vertices of D with a third colour, we have a total rainbow connected colouring
→ →
for D, using 3 colours. Thus, trc(D) ≤ 3. We have trc(D) = 3, and thus both conditions of
→
(iii) hold. Secondly, suppose that either condition in (iii) holds, so that trc(D) = 3. By (1),
we have diam(D) ≤ 2, and (a) implies diam(D) 6= 1, so that diam(D) = 2. Thus, the three
conditions of (ii) also hold.
InTheorem2(c), thereareinfinitelymany examples of digraphsD wheretheconditions of
(ii) and (iii) hold, but the conditions of (i) do not hold. For example, let u be a vertex of the
→ →
directedcycleC3,andletDn bethedigraphonn ≥ 3vertices,obtainedfromC3 byexpanding
u into K ∼= K↔n−2. That is, Dn = (C→3)u→K. Then, we have r→vc(Dn) = sr→vc(Dn) = 1,
→ → → →
diam(D ) = 2, and trc(D ) = strc(D ) = 3, but rc(D ) = src(D ) = 3. Also, there are
n n n n n
infinitely many examples of digraphs D where the conditions of (ii) hold, but the conditions
of (iii) do not hold, as we shall see in the following lemma.
→
Lemma 3. There exist infinitely many digraphs D where diam(D) = 2 and trc(D) =
→
strc(D) = 4.
↔
Proof. We first consider the digraph P10, where P10 is the Petersen graph. See Figure 1(a),
↔ ↔
where we have V(P10) = V(P10) = {ui,vi : 0 ≤ i ≤ 4}. Clearly, we have diam(P10) = 2. We
→ ↔ → ↔ → ↔
will show that trc(P10) = strc(P10) = 4. By (1), it suffices to prove that trc(P10) ≥ 4 and
→ ↔
strc(P10) ≤ 4.
. .
u0 1
u4.........................................................................................................................................................................................................................................................................................................................................................................................................................................................................v.................................................................................................4..................................................................................................................................................................................................................................................................................................................................v.........................................................................................................................................................................3............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................v..........................................................................................................................................................................0.............................................................v...........................................................................................................................................................................2.........................................................................................................................................................................................................................................................................................................................................v..................................................................................................1...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................u1 2...........................................................................................................1.......................................................................................................................................................................................................................................4........................................................................................................................................................3............................................4..2......................................................................................................................................................................................................................................................................................................................................................2................................................................................3..........................................................................................................................................................................................................................................1..................................4.........................................................................................................................................3..................................................................................................................................................................................................................................................................1.....................1.......................................................................................................................................................................................................................................................2.................................................................3......................................................................................................................................................................................................................................................................................................................................................................................................2...........4.....................................2................................................................................................................................................4.................................................................................................................................................................................................................................................1....................................................................................................................................................................3
u3 u2 2 3 2
(a) (b)
Figure 1. (a) The Petersen graph P10; (b) A strongly total rainbow connected colouring for P10.
5
↔
Suppose first that there is a total rainbow connected colouring c of P10, using at most
↔ ↔
three colours, say 1,2,3. Then whenever we have x,y ∈ V(P10) and xy 6∈ A(P10), there must
↔
be a total-rainbow x−y path of length 2 in P10. By considering the unique paths of length
2 between the vertices u0,u2,v1, we may assume that c(u0u1) = c(u2u1) = c(v1u1) = 1,
c(u1u0) = c(u1u2) = c(u1v1) = 2, and c(u1) = 3. Then by considering the unique paths of
length 2 between u1,u3, we must have c(u2u3) = 1, c(u3u2) = 2 and c(u2) = 3. Similarly
by considering the pairs u2,u4, then u3,u0, we must have c(u4u3) = c(u4u0) = 1, c(u3u4) =
c(u0u4) = 2, and c(u3) = c(u4) = 3. But then, we do not have total-rainbow paths between
→ ↔
u1,u4, which is a contradiction. Therefore, trc(P10) ≥ 4.
Now we define a total-colouring of P10 as follows. We assign strongly total rainbow con-
nected colourings for the two cycles u0u1u2u3u4u0 and v0v2v4v1v3v0 with colours 1,2,3, and
then colour 4 to the edges u v , for 0 ≤ i ≤ 4. For example, see Figure 1(b). We can then
i i
extend this to a total-colouring c′ of ↔P10 where two symmetric arcs of ↔P10 both receive the
same colour as the corresponding edge in P10. Then it is easy to check that c′ is a strongly
↔ → ↔
total rainbow connected colouring for P10. Therefore, strc(P10) ≤ 4.
↔
Finallyforeveryn ≥ 11,wecanobtainthedigraphDn onnverticesfromP10 byexpanding
v0 into K ∼=K↔n−9. That is, Dn = (↔P10)v0→→K. Then note that diam(Dn) = 2. We may again
apply the first argument above to obtain trc(D ) ≥ 4. Moreover, we can extend the total-
n
colouring c′ on ↔P10 to a total-colouring c′′ on Dn where c′′(uv) = c′(uv) if uv 6∈ A(K);
′′ ′ ′′ ′
c (uv) = c(v0v) if u ∈ V(K) and v 6∈ V(K); c (uv) = c(uv0) if u 6∈ V(K) and v ∈ V(K);
′′ ′′ ′ ′′
c (uv) = 1 if uv ∈ A(K); c (w) = c(w) if w 6∈ V(K); and c (w) = 1 if w ∈ V(K). Then
′′
it is easy to see that c is a strongly total rainbow connected colouring for D . Indeed, if
n
x,y ∈ V(D ) and xy 6∈ A(D ), then at most one of x,y is in V(K), and x,y correspond
n n
to distinct vertices x′,y′ in ↔P10. Then a total-rainbow x − y geodesic of length 2 in Dn
corresponds to a total-rainbow x′ −y′ geodesic of length 2 in ↔P10. Therefore, st→rc(Dn) ≤ 4,
→ →
and trc(D ) = strc(D )= 4.
n n
We propose the following problem.
→ → →
Problem 4. Among all digraphs D with diameter 2, are the functions rc(D), src(D), trc(D)
→
and strc(D) unbounded?
Next, Alva-Samos and Montellano-Ballesteros [1] showed that for a connected graph G,
→ ↔ → ↔
rc(G) ≤ rc(G) and src(G) ≤ src(G). (2)
Furthermore, for each inequality, there is an infinite family of graphs where equality holds,
and also with the difference between the two parameters unbounded. For example, from
→ ↔ → ↔
[1, 6], we have rc(C ) = rc(C ) = src(C ) = src(C ) = ⌈n⌉ for n ≥ 4. Also, we have
n n n n 2
→ ↔ → ↔
rc(K1,n) = src(K1,n) = 2 and rc(K1,n) = src(K1,n) = n for n ≥ 2, where K1,n is the star
with n edges. On the other hand, for rainbow vertex-connection, Lei et al. [11] showed that
→ ↔ → ↔
rvc(G) = rvc(G) and srvc(G) = srvc(G). (3)
For total rainbow connection, we have the analogous inequalities to (2).
6
Proposition 5. For a connected graph G, we have
→ ↔ → ↔
trc(G) ≤ trc(G) and strc(G) ≤ strc(G). (4)
Proof. Given a (strong) total rainbow connected colouring of G, it is not hard to see that
↔
the total-colouring ofG, obtained by assigning the colour of the edge uv to both arcs uv,vu,
↔
and the colour of the vertex u of G to the corresponding vertex u of G, is a (strong) total
↔
rainbow connected colouring of G.
As in the case for rainbow connection, for each inequality of (4), there is an infinite family
of graphs where equality holds, and also with the difference between the two parameters
→ ↔
unbounded. We simply use the same examples. For n ≥ 3, we have trc(C ) = trc(C ) =
n n
→ ↔
strc(C ) = strc(C ), which we will see in Theorem 11 later. Also, for n ≥ 2, we have
n n
→ ↔ → ↔
trc(K1,n)= strc(K1,n) = 3 and trc(K1,n)= strc(K1,n) = n+1.
For the relationship between the total rainbow connection numbers of a digraph and its
spanning subdigraphs, it is not hard to see that the following holds.
Proposition 6. Let D and H be strongly connected digraphs such that, H is a spanning
→ →
subdigraph of D. Then trc(D) ≤ trc(H).
However, this is not true for the strong total rainbow connection number, as we will see
in the next lemma.
Lemma 7. There are strongly connected digraphs D and H such that, H is a spanning
→ →
subdigraph of D, and strc(D) > strc(H).
. .
.
uvu218v7..21.................................................................................................................1....................................................................................1............................................................................................................................................................................................................................................................................................................................................................................11..1..........2................................................12..............................................................................................................................11..........................................................................................................................................................................................................................................................................................11....................................................43......................................................................................................................................................................................................................................................................................34................................................12..............................................................................................................................................................ww..........................................................................................................12..........11................................................................65...........................................................................................................................................................................................................................................................................................................11.......................................................................................................................................................................................................................................................................................................................................................................................................56.......................................................................................................................................................................................................................................................x........5..........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................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......................11................................................................56..ww...............................................................................................................43.......................................................................................................................................................................43........................................11.................................................................11...........................43............................................................................................................................................................................................................................................................................................................................................................34............................................................................................................................................................................................................................................................................................................11......................................11......................12.........................................12......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................u......v........................................................19430uv34
Figure 2. The digraphs D and H in Lemma 7.
Proof. Let H be the digraph consisting of the solid arcs as in Figure 2, and D bethe digraph
obtainedfromH byaddingthedottedarcxy. ItisnothardtoseethatthecolouringforH as
showninFigure2, is strongly total rainbowconnected, wherefortheeightpairs ofsymmetric
→
arcs in the middle, the two arcs in each pair have the same colour. Thus, strc(H) ≤ 16.
→
Now, we will show that strc(D) ≥ 17. Suppose that we have a strongly total rainbow
connected colouring c for D, using at most 16 colours. Notice that any two cut-vertices of
D must receive distinct colours, and thus without loss of generality, we may assume that
w1,w2,w3,w4,x,y have colours 1,2,3,4,5,6. Next, for 1 ≤ i ≤ 4, we see that the arcs uivi
7
musthavedistinctcolours,anddifferentfromcolours1,2,3,4,5,6. Otherwisewecanfindtwo
vertices such that any geodesic in D connecting them is not total-rainbow. Hence, we may
assumethatc(u v )= i+6,for1 ≤ i≤ 4. Forthesamereason,thearcsw u ,v w andvertices
i i i i i i
u ,v for 1≤ i ≤ 4, and thearcs w x,xw for i = 1,2, and w y,yw for i= 3,4, cannot usethe
i i i i i i
colours 1,2,...,10. Sincetheuniqueu1−v2 geodesic in D is u1v1w1xw2u2v2, wemay assume
that c(v1) = 12, c(v1w1) = 13, c(w1w) = 15, c(xw2) = 16, c(w2u2) = 14, and c(u2) = 11.
Finally, the unique u1 − v3 geodesic in D is u1v1w1xyw3u3v3, and the arcs xy,yw3,w3u3
and internal vertex u3 in this geodesic are not yet coloured. These elements cannot use the
colours 1,2,...,10,12,13,15. Thus the remaining possible colours are 11,14,16, and this is
→
insufficienttomakethegeodesictotal-rainbow. Wehaveacontradiction. Hencestrc(D) ≥ 17
and the result follows.
Our final aim in this section is to compare the rainbow connection parameters. In [10],
Krivelevich and Yuster observed that for rc(G) and rvc(G), we cannot generally find an
upper bound for one of the parameters in terms of the other. Indeed, by taking G = K1,s,
we have rc(G) = s and rvc(G) = 1. On the other hand, let the graph G be constructed as
s
follows. Take s vertex-disjoint triangles and, by designating a vertex from each triangle, add
a complete graph K on the designated vertices. Then rc(G ) ≤ 4 and rvc(G ) = s. In [11],
s s s
→ → → →
similar results were obtained for rc(D) and rvc(D), and for src(D) and srvc(D).
When considering the total connection number in addition, we have the following trivial
inequalities.
→ → →
trc(D)≥ max(rc(D),rvc(D)), (5)
→ → →
strc(D)≥ max(src(D),srvc(D)). (6)
Inthefollowingresult,weseethatthereareinfinitelymanydigraphswheretheinequalities
(5) and (6) are best possible.
Theorem 8.
→ →
(a) There exist infinitely many strongly connected digraphs D with trc(D) = strc(D) =
→ →
rc(D) = src(D) = 3.
→ →
(b) Given s ≥ 13, there exists a strongly connected digraph D with trc(D) = rvc(D) = s.
→ →
(c) Given s≥ 13, there exists a strongly connected digraph D with strc(D) = srvc(D) = s.
Proof. (a) We may simply consider the digraphs D as described after the proof of Theorem
n
→
2. That is, Dn is the digraph on n ≥ 3 vertices, obtained by expanding a vertex of C3 into
↔ → → → →
Kn−2. We have trc(Dn) = strc(Dn)= rc(Dn)= src(Dn) = 3.
(b) For s ≥ 13, let G be the simple graph with s disjoint triangles attached K , as
s s
↔ →
described above. Let D = G . Then by (4), (5) and (3), we have trc(G ) ≥ trc(D ) ≥
s s s s
→
rvc(D ) = rvc(G ) = s. In [15], Theorem 11, it was proved that trc(G ) = s. Thus we have
s s s
→ →
trc(D ) = rvc(D ) = s.
s s
(c) For s ≥ 13, we construct the simple graph H as follows. First, we take the graph
s
8
Gs, and let u1,...,us be the vertices of the Ks, and the remaining vertices are vi,wi, where
uiviwi is a triangle, for 1 ≤ i ≤ s. We then add new vertices z1,...,zs, and connect the
edges uizi,ui+1zi,vizi,wizi+4, for 1 ≤ i≤ s. Throughout, all indices are taken modulo s. See
Figure 3 for the case s = 13.
. .
. v1 w1 .
vww1111v0.vz1................................................1......................................................1................1...................z................................................................................2...................................................0.............0...................................................1......................................................w........................................................................................................................1............................................................................................................................................................w.........................................................................................................................................................................................................................................................................1.............................................................................................................................z.............................................................................................................................................9........................................2..................................................................................................................................................................................................................9............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................u...v...............................................................................................................................................................................................................................................................................................................................................................u...................................................................................................................1..............................................................................1..................................................................................................z......................................................................................................................................................................3.....................................1......................................1....................................................v................................................................................u.................................................1....................................................................................................................................................................0...................................................................................................................................................................................9.............................................2.................................................................................................................................................1.............................................................................................................................................................................................................................................................................................................................................................................................................................................................2.................................................................................................................................................................................................................................u.........................................................................................................................................................................................................................................................z........................................................................................................................................................................................................................................................................................................................9................w.................................................................................w..............................................8............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................1.....................................................................................................................................8..............................................................................................................................................................u................................................................................................................................................3..................................................................................................................................................................................................................................................................................................................................1....................................................................................................................................................................................................................................................................................................z..............................3........................................................................................................................................................................................................................................................................................................................................1.............................u................................................................................................................................................................................................................................................3.......................................................v......................................................................................................................8......................................................................................................................................................................................................................................8..........................................................................................................................................................................................................................................................................................................K................................................................................................................u.z...................................................................................................................................................................................................................................................................................................................7...............................................................................1..................................................................................1......................................................................................................................................................................................................................................................................................................................3...........................................................................................w............................................................................................................................................................................................................................................................................................................................................................................................................................................................................u..............................................................................................................7.................................................................................................................................................................................................................................................................................................................................................................................................................7..................................................................................z...............................................................................................................................................................................................................................................................................................................................................1..........................................................................................................................................................................................................................................................................................................................................................................................................................u.....................................................................................................................................................v........................................................................................................................v...............................................................................................................................................................................................................................................2.................................................................................2.....................................................................................................................7.........................z.........................................................................................................................................................................................................................................................................................u.................................................................................................................................................................6...................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................6...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................u.................................................................................................................................................................................................................................ww...........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................3..................................................u.........................................................z...................................................................................................................................26.............................................................................................................................................................................u............................................................2.........................5......................................................................................................................................................................................................................................................................................................................................4...............................................................................................................................................................................................................................................................................................................................................................................v...........................................................................................................................................................z....................................................................................................................................................................................................................................................................................................................3..................................................................................................................................v....................................5..........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................6................................................................................................................................................................................z..................................................................................................................................................................................................................................................................................................3...................................z............................................ww................................................................................................................................................................................................4.....................................................................................................................................................35............................................................v..............vw454
Figure 3. The graph Hs when s=13. The polygon with dotted lines represents the copy of Ks.
↔ → →
Now let D =H . Then by (4), (6) and (3), we have strc(H ) ≥ strc(D ) ≥ srvc(D ) =
s s s s s
srvc(H ). Thus it suffices to prove that srvc(H ) ≥ s and strc(H ) ≤ s, which would imply
s s s
→ →
strc(D )= srvc(D )= s.
s s
We first prove that srvc(H ) ≥ s. Suppose that we have a vertex-colouring of H , using
s s
fewer than s colours. Then some two vertices of u1,...,us have the same colour. We may
assume that u1 and ui have the same colour, for some 2 ≤ i ≤ s. If i 6∈ {5,6,s−4,s−3},
then the unique w1−wi geodesic is w1u1uiwi. If i∈ {5,6}, then the unique v1−wi geodesic
is v1u1uiwi. If i∈ {s−4,s−3}, then the unique w1−vi geodesic is w1u1uivi. In each case,
we have two vertices in H such that there is no vertex-rainbow geodesic connecting them.
s
Thus srvc(H ) ≥ s.
s
Now we prove that strc(H ) ≤ s. We define a total-colouring c of H , using the colours
s s
1,...,s, as follows. For 1 ≤ i ≤ s, let c(ui) = c(wizi+4) = i, c(uivi) = i+1, c(vi) = i+2,
c(viwi) = c(ui+1zi) = i+3, c(wi) = c(zi) = i+4, c(wiui) = c(uizi) = c(vizi) = i+5, all
modulos. For i 6= j, let c(u u ) bea colour not in {i,...,i+5}∪{j,...,j+5}. Such a colour
i j
exists since s ≥ 13. We will show that c is a strongly total rainbow connected colouring
for H , which implies that strc(H ) ≤ s. Let x,y ∈ V(H ). We show that there always
s s s
exists a total-rainbow x−y geodesic P. If at least one of x,y belongs to {u1,...,us}, or if
x,y ∈ {v ,w ,z } for some i, then clearly such an x−y geodesic exists, with length at most
i i i
2. Thus, it suffices to consider x ∈ {v1,w1,z1} and the six cases as shown in the following
table, where 2 ≤ i ≤s in each case. In each case, we have a desired x−y geodesic P, where t
9
is some colour different from the remaining four colours. For example, in the very first case,
we have t 6∈{1,2,i,i+1}.
x y P Colours of P
v1u1uivi if i6∈ {2,s} 2,1,t,i,i+1
v1 vi v1z1u2v2 if i= 2 6,5,4,2,3
v1u1zsvs if i= s 2,1,3,4,5
v1u1uiwi if i6∈ {2,s−4,s−3} 2,1,t,i,i+5
v1z1u2w2 if i= 2 6,5,4,2,7
v1 wi
v1u1zsws−4 if i= s−4 2,1,3,4,s−4
v1z1ws−3 if i= s−3 6,5,s−3
v1u1uizi if i6∈ {2,5,s−4,s−3,s} 2,1,t,i,i+5
v1u1ui+1zi if i∈ {2,s−4,s−3} 2,1,t,i+1,i+3
v1 zi
v1w1z5 if i= 5 4,5,1
v1u1zs if i= s 2,1,3
w1u1uiwi if i6∈ {6,s−4} 6,1,t,i,i+5
w1 wi w1z5u6w6 if i= 6 1,9,8,6,11
w1u1zsws−4 if i= s−4 6,1,3,4,s−4
w1u1uizi if i6∈ {5,6,s−4,s} 6,1,t,i,i+5
w1z5 if i= 5 1
w1 zi
w1u1ui+1zi if i∈ {6,s−4} 6,1,t,i+1,i+3
w1u1zs if i= s 6,1,3
z1u1uizi if i6∈ {2,6,s−4,s} 6,1,t,i,i+5
z1u2z2 if i= 2 4,2,7
z1 zi
z1u1ui+1zi if i∈ {6,s−4} 6,1,t,i+1,i+3
z1u1zs if i= s 6,1,3
The proof is thus complete.
We may also consider how far from equality we can be in (5) and (6). In the following
→
result, we see that there is an infinite family of digraphs D such that trc(D) is unbounded on
→ → →
D, while rc(D) is bounded. Similar results also hold for trc(D) in comparison with rvc(D),
→ → →
and for strc(D) in comparison with each of src(D) and srvc(D).
Theorem 9.
→ →
(a) Givens ≥ 2, there existsastrongly connected digraph D suchthatstrc(D) ≥trc(D) ≥ s
→ →
and rc(D) = src(D) = 3.
→ →
(b) Givens ≥ 4, there existsastrongly connected digraph D suchthattrc(D) = strc(D) ≥ s
→ →
and rvc(D) = srvc(D)= 3.
Proof. (a) Let F be the simple graph consisting of K with a pendent edge attached to
s s
↔
each of the s vertices of Ks, and Ds = Fs. Let u1,...,us be the vertices of the Ks, and
v1,...,vs be the pendent vertices, where uivi ∈ E(Fs) for 1 ≤ i ≤ s. Then in any total
rainbow connected colouring of D , the vertices u must receive distinct colours, and thus
s i
10
