Equitable neighbour-sum-distinguishing edge and total colourings

0.26 MB·

by  Olivier Baudon

Equitable neighbour-sum-distinguishing
edge and total colourings
Olivier Baudonb,c, Monika Pil´sniaka, Jakub Przybyl oa, Mohammed Senhajib,c, E´ric
Sopenab,c, Mariusz Wo´zniaka,1
aAGH University of Science and Technology, al. A. Mickiewicza30, 30-059 Krakow, Poland
bUniv. Bordeaux, LaBRI, UMR5800, F-33400 Talence, France.
7 cCNRS, LaBRI, UMR5800, F-33400 Talence, France.
1
0
2

n
a
Abstract
J

7 Withany(notnecessarilyproper)edgek-colouringγ :E(G)−→{1,...,k}ofagraphG,
1 one can associate a vertex colouring σ given by σ (v) = γ(e). A neighbour-sum-
γ γ e∋v
distinguishing edge k-colouring is an edge colouring whose associated vertex colouring
] P
M is proper. The neighbour-sum-distinguishing index of a graph G is then the smallest
k for which G admits a neighbour-sum-distinguishing edge k-colouring. These notions
D
naturally extends to total colourings of graphs that assign colours to both vertices and
.
s edges.
c We study in this paper equitable neighbour-sum-distinguishing edge colourings and
[
total colourings, that is colourings γ for which the number of elements in any two

1 colour classes of γ differ by at most one. We determine the equitable neighbour-sum-
v
distinguishing index of complete graphs, complete bipartite graphs and forests, and the
8
equitable neighbour-sum-distinguishing total chromatic number of complete graphs and
4
6 bipartite graphs.
4
Keywords: Equitable colouring, Neighbour-sum-distinguishing edge colouring,
0
. Neighbour-sum-distinguishing total colouring
1
0
7
1 1. Introduction and statement of results
:
v
We consider undirected simple graphs and denote by V(G) and E(G) the sets of
i
X vertices and edges of a graph G, respectively. We denote by deg (u), or simply deg(u)
G
r whenever the graph G is clear from the context, the degree of a vertex u in G.
a
Anedge k-colouring, k ≥1, ofa graphGis a mapping γ :E(G)−→{1,...,k}. Note
thatsuchedgecolouringisnotnecessarilyproper. Eachedgecolouringnaturallyinduces
a vertex colouring σ (v) given by
γ
σ (v)= γ(e)
γ
e∋v
X
for every v ∈ V(G). We will write σ(v) instead of σ (v) whenever the edge colouring
γ
γ is clear from the context. We call the value γ(e), e ∈ E(G), the colour of e and the
value σ(v), v ∈ V(G), the sum at v. For an edge uv ∈ E(G), we will say that u and v
Preprint submitted to Elsevier January 18, 2017
are in conflict whenever σ(u) = σ(v). A neighbour-sum-distinguishing edge k-colouring
of G (edge k-nsd-colouring for short) is an edge k-colouring γ such that for every edge
uv ∈E(G), σ (u)6=σ (v). Clearly, a graph admits an edge nsd-colouring if and only if
γ γ
it has no isolated edge. The smallest k for which G admits an edge k-nsd-colouring is
the nsd-index of G, denoted χe(G).
Σ
Atotalk-colouring,k≥1,ofagraphGisamappingγ :V(G)∪E(G)−→{1,...,k}.
t
Again, note that we do not require a total colouring to be proper. Similarly as above,
each total colouring naturally induces a vertex colouring σT(v) given by
γt
σT(v)=γ (v)+ γ (e)
γt t t
e∋v
X
for every v ∈ V(G). We will write σT(v) instead of σT(v) whenever the total colouring
γt
γ is clear from the context. We call the value γ (e), e ∈ E(G), the colour of e, the
t t
value γ (v), v ∈V(G), the colour of v and the value σT(u), u∈V(G), the sum at u. A
t
total k-nsd-colouring of G is a total k-colouring γ such that for every edge uv ∈E(G),
t
σT(u) 6= σT(v). Clearly, every graph admits a total nsd-colouring. The smallest k for
γt γt
which G admits a total k-nsd-colouringis the total nsd-chromatic number of G, denoted
χt (G).
Σ
The study of edge nsd-colourings of graphs was initiated by Karon´ski, L uczak and
Thomason [9]. They conjectured that every graph with no isolated edge admits an edge
nsd-colouring with three colours: 1,2,3, and proved it for 3-colourable graphs. Despite
many efforts to tackle this conjecture, see e.g. [1, 2, 15], it is still an open question. The
best knownresultis due to Kalkowski,Karon´skiand Pfender [8], who provedthat every
graph with no isolated edge admits an edge nsd-colouring with five colours, 1,...,5.
Totalnsd-colouringsofgraphswereintroducedin[14]bytwooftheauthors,whocon-
jecturedthateverygraphadmitsatotal2-nsd-colouring,andprovedthis factforseveral
graph families including in particular 3-colourable, 4-regular and complete graphs. The
best general upper bound is in this case due to Kalkowski, who proved in [7] sufficiency
of integers 1,2,3 for constructing a total nsd-colouring of any graph.
A colouring γ with the elements of {1,...,k} is equitable if the numbers of elements
in any two colour classes differ by at most one, that is −1 ≤ |γ−1(i)|−|γ−1(j)| ≤ 1
for any two colours i and j with i,j ∈ {1,...,k}. In 1964, Erdo¨s [5] conjectured that
everygraphwithmaximumdegreearmostr admits anequitable(r+1)-colouring. This
conjecture wasprovedin1970by HajnalandSzemer´edi[6]andis nowknownasHajnal-
Szemer´ediTheorem. Ashorterproofofthistheoremwasgivenin2008byKiersteadand
Kostochka [11]. For a recent survey on equitable colourings,see [12].
In this paper, we study equitable edge and total nsd-colourings. We will denote
by χe(G) the equitable nsd-index of G, that is the smallest k for which G admits an
Σ
equitable edge k-nsd-colouring. Similarly, we will denote by χt (G) the equitable total
Σ
nsd-chromatic number ofG, thatis the smallestk for whichG admits anequitable total
k-nsd-colouring.
Ifγ isanedgek-nsd-colouringofagraphG,thenthemappingγ definedbyγ (v)=1
t t
foreveryv ∈V(G)andγ (e)=γ(e)foreverye∈E(G)is clearlyatotalk-nsd-colouring
t
of G since σ (v) = σ (v)+1 for every vertex v ∈ V(G). Hence, χt (G) ≤ χe(G) for
γt γ Σ Σ
every graph G. The same relation holds for equitable nsd-colourings:
2
Proposition 1. For every graph G without isolated edges, χt (G)≤χe(G).
Σ Σ
Proof. Let γ be an equitable edge k-nsd-colouring of G. We will extend γ to an
equitable total k-nsd-colouring γ of G with γ (e) = γ(e) for every edge e ∈ E(G).
t t
We thus need to extend γ to vertices in such a way that no two adjacent vertices are
in conflict and the colouring remains equitable. We first order the vertices of G as
v ,v ,...,v , n=|V(G)|, in such a way that
1 2 n
σ (v )≤σ (v )≤···≤σ (v ). (1)
γ 1 γ 2 γ n
Let q = |γ−1(i)| be the number of edges with colour i. Moreover, let q and r be non-
i
negative integers such that r <k and |V(G)|+|E(G)|=qk+r. For γ to be equitable,
t
we then must haver colourclassesof orderq+1 and k−r colour classesof orderq. Let
C be any subset of k−r colours from {1,...,k} such that q ≤q for every i∈C , and
q i q
C = {1,...,k}\C . We will then colour q′ = q−q vertices with colour i for each
q+1 q i i
i∈C and q′ =q+1−q vertices with colour j for each j ∈C . In order to produce
q j j q+1
a total nsd-colouring, we will colour the vertices according to the above defined order,
and assign the colour 1 to the first q′ vertices, then colour 2 to the next q′ vertices and
1 2
so one. More formally, we let
j
γ (v )=min j q′ ≥i . (2)
t i p
( )
(cid:12) Xp=1
(cid:12)
For every edge v v , i<j, we then have (cid:12)
i j
σ (v )=σ (v )+γ (v )<σ (v )+γ (v )=σ (v )
γt i γ i t i γ j t j γt j
since σ (v )<σ (v ) (by (1), as γ is neighbour-sum-distinguishing) and γ (v )≤γ (v )
γ i γ j t i t j
(by (2)). The total colouring γ is therefore an equitable total nsd-colouring, and thus
t
χt (G)≤χe(G). (cid:3)
Σ Σ
It is known that for every graph G with no isolated edge, the nsd-index of G is 3
if G is a complete graph, see e.g. [4], 2 if G is a complete bipartite graph [13] and at
most 2 if G is a forest [10]. Observe that we necessarily have χt (G) ≥ 2, and thus
Σ
χe(G) ≥ 2, whenever two adjacent vertices in G have the same degree (in particular
Σ
when G is regular).
Concerningtheequitablensd-indexofthesegraphclasses,wewillprovethefollowing:
Theorem 2. For every complete graph K with n ≥ 3, n 6= 4, χe(K ) = 3, while
n Σ n
χe(K )=4.
Σ 4
Theorem 3. For every complete bipartite graph K with m = n = 2 or m = n ≥ 4,
m,n
χe(K )=2, while χe(K )=3 and χe(K )=1 if 1≤m<n.
Σ m,n Σ 3,3 Σ m,n
Theorem 4. For every forest F with no isolated edge, χe(F)≤2.
Σ
Itwasprovedin[14]thatthe totalnsd-chromaticnumberofGis2ifGisacomplete
graph of order n ≥ 2, and at most 2 if G is bipartite. We prove that the same result
holdsfortheequitabletotalnsd-chromaticnumberofbipartitegraphs,andthatweneed
one more colour to produce an equitable total nsd-colouring of the complete graph K
n
whenever n≥3:
3
Theorem 5. For every bipartite graph G, χt (G)≤2.
Σ
Theorem 6. For every complete graph K with n≥3, χt (K )=3, while χt (K )=2.
n Σ n Σ 2
The proofs of Theorems 2 to 6 are given in the next sections.
2. Proof of Theorem 2
We proveinthis sectionthatχe(K )=3foreveryn≥3, n6=4,andthatχe(K )=
Σ n Σ 4
4. Recall that since K is regular, we have χe(K ) ≥ 2. We first introduce some
n Σ n
definitions and preliminary results.
For any edge 3-colouring γ of a graph G, we denote by E (i), 1 ≤ i ≤ 3, the set of
γ
edges of G with colour i. For every vertex v of G, we denote by d (v) (or simply d (v)
γ,i i
when γ is clear from the context), 1 ≤ i ≤ 3, the number of edges of E (i) that are
γ
incident with v.
Let σ denote the vertex colouring of G induced by γ, and σ denote the mean value
of σ on V(G), that is
1
σ = σ(v).
|V(G)|
v∈XV(G)
The σ-deviation of a vertex v is the value µ (v) = σ(v)−σ. For regular graphs, the
σ
deviation can be computed as follows:
Lemma 7. LetGbea d-regular graph, γ bean edge 3-colouring ofG andσ be thevertex
colouring of G induced by γ. We then have σ(v)=d (v)−d (v)+2d for every vertex
γ,3 γ,1
v in G. Moreover, if |E (1)| = |E (3)|, then σ = 2d and µ (v) = d (v)−d (v) for
γ γ σ γ,3 γ,1
every vertex v in G.
Proof. We have
σ(v) = d (v)+2d (v)+3d (v)
γ,1 γ,2 γ,3
= d (v)−d (v)+2(d (v)+d (v)+d (v)) .
γ,3 γ,1 γ,1 γ,2 γ,3
= d (v)−d (v)+2d.
γ,3 γ,1
Suppose now that |E (1)|=|E (3)|. We then have
γ γ
σ(v) = (d (v)−d (v)+2d (v))
v∈V(G) v∈V(G) γ,3 γ,1 G
= 2|E (3)|−2|E (1)|+2d|V(G)|
γ γ
P P
= 2d|V(G)|.
This gives σ = 2d. Moreover, since σ(v) = d (v) − d (v) + 2d, we get µ (v) =
γ,3 γ,1 σ
σ(v)−σ =d (v)−d (v). (cid:3)
γ,3 γ,1
We say that an edge 3-colouring γ of G is good if the following conditions hold:
1. |E (1)|=|E (3)|,
γ γ
2. for every vertex v in G,
|V(G)| |V(G)|
− ≤µ (v)≤ ,
σ
2 2
(cid:22) (cid:23) (cid:22) (cid:23)
4
3. there exist two vertices wγ and wγ in G such that
min max
|V(G)|
µ (wγ )=−
σ min 2
(cid:22) (cid:23)
and
|V(G)|
µ (wγ )= .
σ max 2
(cid:22) (cid:23)
where σ denotes the vertex colouring of G induced by γ.
Lemma 8. For every integer n≥3, if γ is an equitable edge 3-colouring of the complete
graph K such that |E (1)|=|E (3)|, then either
n γ γ
|E (2)|=|E (1)|=|E (3)|
γ γ γ
or
|E (2)|=|E (1)|+1=|E (3)|+1.
γ γ γ
Proof. Since |E(K )| = n(n−1), if n ≡ 0 or 1 (mod 3), then |E(K )| ≡ 0 (mod 3),
n 2 n
which implies |E (2)| = |E (1)| = |E (3)|. On the other hand, if n ≡ 2 (mod 3), then
γ γ γ
|E(K )|≡1 (mod 3), which implies |E (2)|=|E (1)|+1=|E (3)|+1. (cid:3)
n γ γ γ
We are now able to prove Theorem 2.
Proof (of Theorem 2). We first prove a series of claims concerning equitable edge
nsd-colourings of K for small values of n.
n
Claim 1. For every integer n≥3, χe(K )≥3.
Σ n
Proof. Assume to the contrary that γ is an equitable edge 2-nsd-colouring of K and
n
let σ be the vertex colouring induced by γ. We then have n−1 ≤ σ(v) ≤ 2(n−1) for
every vertex v of K , hence, since we need n distinct values for n vertices, there exist
n
two vertices u and v with σ(u) = n−1 and σ(v) = 2(n−1), in contradiction with the
colour of the edge uv. (cid:3)
Claim 2. For every integer n ∈ {3,5,6}, there is a good equitable edge 3-nsd-colouring
of K .
n
Proof. A goodequitable edge 3-nsd-colouringofK is obtainedby colouringthe edges
3
of K with colours 1, 2 and 3, respectively.
3
A good equitable edge 3-nsd-colouring of K is depicted in Figure 1, together with
5
its matrix representation. The value in row i and column j is the colour of the edge ij.
The sum at vertex i is given at the end of row i.
The matrix representationof a goodequitable edge 3-nsd-colouringof K is given in
6
Figure 2. (cid:3)
Claim 3. χe(K )=4.
Σ 4
5
7
2 1
1 − 1 1 1 3 6
8 6 1 − 2 2 2 7
 
2 2 1 2 − 3 2 8
1 2 3 − 3 9
2 1  
3 3 3 2 2 3 − 10
 
 
3
9 10
Figure1: Anequitableedge3-nsd-colouringofK5 anditsmatrixrepresentation.
− 1 1 1 1 3 7
1 − 1 2 2 2 8
 
1 1 − 2 3 2 9
1 2 2 − 3 3 11
 
1 2 3 3 − 3 12
 
3 2 2 3 3 − 13
 
 
Figure2: Matrixrepresentationofanequitableedge3-nsd-colouringofK6.
Proof. Assume first that γ is an equitable edge 3-nsd-colouring of K and let σ be
4
the vertex colouring induced by γ. Since |E(K )| = 6, we necessarily have |E (1)| =
4 γ
|E (2)| = |E (3)| = 2 which implies d (v) ≤ 2 for every v ∈ V(K ) and every i,
γ γ γ,i 4
1 ≤ i ≤ 3. Hence, 4 ≤ σ(v) ≤ 8 for every v ∈ V(K ). On the other hand, we have
4
σ(v)=2 γ(e)=24.
v∈V(K4) e∈E(K4)
Let V(K )={v ,v ,v ,v }. To get a total sum of 24, we thus have, without loss of
P 4 P1 2 3 4
generality,σ(v )=4,σ(v )=5,σ(v )=7andσ(v )=8. Theedgesincidentwithv are
1 2 3 4 1
thus coloured 1, 1 and 2, while the edges incident with v are coloured 3, 3 and 2. This
4
implies γ(v v ) = 2 and thus γ(v v ) = 1, γ(v v ) = 1, γ(v v ) = 3 and γ(v v ) = 3,
1 4 1 2 1 3 2 4 3 4
hence γ(v v )=2. Thus, we finally get σ(v )=σ(v )=6, a contradiction.
2 3 2 3
Hence, χe(K ) > 3. An equitable edge 4-nsd-colouring of K is given in Figure 3,
Σ 4 4
together with its matrix representation. (cid:3)
We will now prove that χe(K ) = 3 for every n ≥ 7, by induction on n. More
Σ n
precisely, we will show that for every n ≥ 5 any good equitable edge 3-nsd-colouring of
K can be extended to a good equitable edge 3-nsd-colouring of K . Together with
n n+2
claims 1 to 3, this will complete the proof.
Let γ be a good equitable edge 3-nsd-colouring of K , n≥5. Suppose that K is
n n+2
obtained from K by adding two new vertices u and v. Let S ⊆V(K ) be any fixed set
n n
of⌊n⌋+1=⌊n+2⌋verticesandS =V(K )\S (wethushaveV(K )=S∪S∪{u,v}).
2 2 n n+2
We first define an edge 3-colouring γ of K as follows:
0 n+2
6
6
2 − 3 1 2 6
3 − 4 1 8
1 3  
1 4 − 2 7
5 2 1 2 − 5
 
 
2 1
7 8
4
Figure3: Anequitableedge4-nsd-colouringofK4 anditsmatrixrepresentation.
•
•
.
.
.
•
S
u • • v
S
•
.
.
.
•
•
Figure4: Theedge3-colouringγ0 ofKn+2.
1. γ (xy)=γ(xy) for every edge xy with x,y ∈S∪S,
0
2. γ (uv)=2,
0
3. for every vertex x∈S, γ (ux)=1 and γ (vx)=3,
0 0
4. for every vertex y ∈S, γ (uy)=γ (vy)=2.
0 0
Theedge3-colouringγ isdepictedonFigure4(dashed,thinandthickedgesrepresent
0
edges with colour 1, 2 and 3, respectively). This colouring is not equitable but is indeed
good and neighbour-sum-distinguishing:
Claim 4. The edge 3-colouring γ is a good edge 3-nsd-colouring of K .
0 n+2
Proof. Recall that γ is a good equitable edge 3-nsd-colouring of K . We denote by σ
n
and σ the vertex colourings induced by γ and γ , respectively.
0 0
For eachvertex x∈S∪S, the two edges ux andvx are either assignedcolours1 and
3orbothassignedcolour2byγ . Therefore,foreveryvertexx∈S∪S, µ (x)=µ (x),
0 σ0 σ
7
µ (wγ )=⌊n⌋ µ (v)=⌊n⌋+1 µ (wγ1 )=⌊n⌋+1 µ (v)=⌊n⌋
σ0 max 2 σ0 2 σ1 max 2 σ1 2
• • • •
−→
• • • •
µσ0(u)=−⌊n2⌋−1 µσ0(wmγin)=−⌊n2⌋ µσ1(u)=−⌊n2⌋ µσ1(wmγ1in)=−⌊n2⌋−1
Figure5: Recolouringoftype1.
yi v yi v
• • • •
−→
• • • •
u xi u xi
Figure6: Recolouringoftype2.
whichimplies−⌊n⌋≤µ (x)≤⌊n⌋. Ontheotherhand,µ (u)=−(⌊n⌋+1)=−⌊n+2⌋
2 σ0 2 σ0 2 2
and µ (v)= ⌊n⌋+1 =⌊n+2⌋. Hence, all vertices of K are assigned distinct values
σ0 2 2 n+2
by µ . Since |E (1)| = |E (1)| + ⌊n+2⌋ and |E (3)| = |E (3)| + ⌊n+2⌋, so that
σ0 γ0 γ 2 γ0 γ 2
|E (1)|=|E (3)|, we thus get by Lemma 7 that γ is a good edge 3-nsd-colouring. (cid:3)
γ0 γ0 0
Whileconstructingγ fromγ,weadded⌊n+2⌋edgeswithcolour1,⌊n+2⌋edgeswith
0 2 2
colour 3 and 2n+1−2⌊n+2⌋ edges with colour 2. The edge colouring γ is thus not
2 0
necessarilyequitable. Wewillthenmodifytheedgecolouringγ inordertoobtainagood
0
equitableedgecolouringγ . Inordertodothat,weneedtorecolourby1or3someedges
1
which are coloured by 2, say p = 2q such edges, leading to an edge colouring γ (with
1
induced vertex colouring σ ) such that |E (1)| = |E (1)|+q, |E (3)| = |E (3)|+q
1 γ1 γ0 γ1 γ0
and|E (2)|=|E (2)|−2q witheither |E (2)|=|E (1)| or|E (2)|=|E (1)|+1 by
γ1 γ0 γ1 γ1 γ1 γ1
Lemma 8.
Theedgecolouringγ willbeproducedusingtwotypesofrecolourings,bothinvolving
1
edges incident with u or v, described as follows:
• Recolouring of type 1 (see Figure 5): Let wγ and wγ denote the (unique)
min max
two vertices such that µ (wγ ) = −⌊n⌋ (= µ (wγ )) and µ (wγ ) = ⌊n⌋
σ0 min 2 σ min σ0 max 2
(= µ (wγ )). If uwγ vwγ is a 2-monochromatic 4-cycle, then recolour with
σ max min max
1 the edge vwγ and recolour with 3 the edge uwγ . Note that the deviations
min max
of u and wγ , and of v and wγ have been switched, so that wγ1 = wγ and
min max min min
8
wγ1 =wγ .
max max
• Recolouring of type 2 (see Figure 6): If the set of pairs of vertices {(x ,y )} ,
i i 1≤i≤k
k ≥1,is suchthatux vy is a2-monochromatic4-cycleforeveryi,1≤i≤k,then
i i
recolour with 1 all edges ux and vy and recolour with 3 all edges uy and vx .
i i i i
Note that the deviation of any of these 2k+2 vertices remains unchanged.
Recall that we need to recolour p = 2q edges which are coloured by 2, q of them
with colour 1 and the q others with colour 3. If q = 1, since n ≥ 5, we can ensure that
the chosen set S contains none of the vertices wγ and wγ . By doing so and then
min max
applying the recolouring of type 1, we obtain an edge colouring γ such that:
1
1. |E (1)|=|E (3)| and either |E (2)|=|E (1)| or |E (2)|=|E (1)|+1,
γ1 γ1 γ1 γ1 γ1 γ1
2. for every vertex x∈V(K ), −⌊n+2⌋≤σ (x)≤⌊n+2⌋,
n+2 2 1 2
3. wγ1 =wγ and wγ1 =wγ .
min min max max
Hence, γ is an equitable good edge 3-nsd-colouring of K and we are done.
1 n+2
Assume from now on that q ≥ 2. Since γ is an equitable good edge 3-nsd-colouring
of K , we know by Lemma 8 that
n
|E (1)|=|E (3)|=r, and r ≤|E (2)|≤r+1,
γ γ γ
with r = ⌊n(n−1)⌋. As observed before, considering the way the edge colouring γ has
6 0
been constructed (see Figure 4), we also have
n+2
|E (1)|=|E (3)|=r+ ,
γ0 γ0 2
(cid:22) (cid:23)
and
n+2 n+2
r+2n+1−2 ≤|E (2)|≤r+2n+1−2 +1.
2 γ0 2
(cid:22) (cid:23) (cid:22) (cid:23)
Again by Lemma 8, in order to be an equitable good edge 3-nsd-colouring of K , the
n+2
edge colouring γ must be such that
1
n+2
|E (1)|=|E (3)|=r+ +q,
γ1 γ1 2
(cid:22) (cid:23)
and
n+2 n+2
r+ +q ≤|E (2)|≤r+ +q+1. (3)
2 γ1 2
(cid:22) (cid:23) (cid:22) (cid:23)
On the other hand, since γ has been obtained by recolouring 2q edges which were
1
coloured with colour 2 by γ , we also have
0
n+2 n+2
r+2n+1−2 −2q≤|E (2)|≤r+2n+1−2 −2q+1. (4)
2 γ1 2
(cid:22) (cid:23) (cid:22) (cid:23)
Combining (3) and (4), we get
n+2 n+2
r+ +q ≤r+2n+1−2 −2q+1,
2 2
(cid:22) (cid:23) (cid:22) (cid:23)
9
which gives
n+2
3q≤2n+1−3 +1.
2
(cid:22) (cid:23)
Since q ≥2, we thus necessarily have n=11 or n≥13. Moreover,we also get
n+2 n+2
|S|=n− ≥3q−n+2 −2≥3q−1≥q+3≥5. (5)
2 2
(cid:22) (cid:23) (cid:22) (cid:23)
The edge colouring γ is then obtained as follows, depending on the parity of q.
1
1. q =2t, t≥1.
WechooseanysetoftpairsofverticesX ={(x ,y )} inS. Sinceux vy isa2-
i i 1≤i≤t i i
monochromatic4-cycleforeveryi,1≤i≤t,wecanapplytherecolouringoftype2
tothesetX,sothat|E (1)|=|E (3)|=|E (1)|+qand|E (2)|=|E (2)|−2q.
γ1 γ1 γ0 γ1 γ0
The so-obtained edge colouring γ is thus an equitable good edge 3-nsd-colouring
1
of K .
n+2
2. q =2t+1, t≥1.
WefirstchooseanysetoftpairsofverticesX ={(x ,y )} inS\{wγ ,wγ }
i i 1≤i≤t−1 min max
(this is possible since, by (5), |S| ≥ q+3). Since ux vy is a 2-monochromatic 4-
i i
cycle for every i, 1 ≤ i ≤ t, we can apply the recolouring of type 2 to the set X,
so that we have |E (1)|+q−1 edges coloured by 1 (resp. by 3). We then apply
γ0
the recolouringof type 1, since uwγ vwγ is still a 2-monochromatic4-cycle, so
min max
that |E (1)|=|E (3)|=|E (1)|+q and |E (2)|=|E (2)|−2q.
γ1 γ1 γ0 γ1 γ0
Therefore, the so-obtained edge colouring γ is an equitable good edge 3-nsd-
1
colouring of K .
n+2
This concludes the proof of Theorem 2. (cid:3)
3. Proof of Theorem 3
We prove in this section that χe(K ) = 2 whenever m = n = 2 or m = n ≥ 4,
Σ m,n
χe(K )=3 and χe(K )=1 if 1≤m<n. As in the previous section, we denote by
Σ 3,3 Σ m,n
E (i) the set of edges that are assigned colour i by the edge colouring γ.
γ
Proof (of Theorem 3). If 1 ≤ m < n, then adjacent vertices have distinct degrees,
and hence colouring all edges with colour 1 gives an equitable edge 1-nsd-colouring of
K .
m,n
If m = n, then K is regular, and therefore, χe(K ) ≥ 2. Suppose first that
m,n Σ m,n
m=n=3 and let V ∪V′ denotes the bipartition of V(K ), with V ={v ,v ,v } and
3,3 1 2 3
V′ ={v′,v′,v′}. We firstclaimthatχe(K )>2. Assume to the contrarythatγ is an
1 2 3 Σ 3,3
equitable edge 2-nsd-colouringof K and let σ denote the vertex colouring induced by
3,3
γ. Since |E(K )|=9, we necessarily have
3,3
{|E (1)|,|E (2)|}={4,5}. (6)
γ γ
Moreover, since 3 ≤ σ(v) ≤ 6 for every vertex v ∈ V ∪V′, we get without loss of
generality either σ(v ) = σ(v ) = σ(v ), or σ(v ) = σ(v ) 6= σ(v′) = σ(v′). In the first
1 2 3 1 2 1 2
10