Sequential edge-coloring on the subset of vertices of almost regular graphs

0.07 MB·

by  Petros A. Petrosyan

Sequential edge-coloring on the subset
of vertices of almost regular graphs
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Petros A. Petrosyan
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a Department of Informatics and Applied Mathematics,
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Yerevan State University, 0025, Armenia

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Institute for Informatics and Automation Problems,

] National Academy of Sciences, 0014, Armenia
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E-mail: pet [email protected]
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h. January 7, 2014
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Abstract
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LetGbeagraphandR⊆V(G). Aproperedge-coloringofagraphG
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withcolors1,...,tiscalledanR-sequentialt-coloringiftheedgesincident
v
6 toeachvertexv∈Rarecoloredbythecolors1,...,dG(v),wheredG(v)is
3 thedegreeofthevertexv inG. Inthisnote,weshowthatifGisagraph
8 with ∆(G)−δ(G) ≤ 1 and χ′(G) = ∆(G) = r (r ≥ 3), then G has an
0 R-sequential r-coloring with |R| ≥ (r−1)nr+n , where n = |V(G)| and
l r m
.
1 nr = |{v ∈ V(G) : dG(v) = r}|. As a corollary, we obtain the following
0 result: if G is a graph with ∆(G)−δ(G) ≤ 1 and χ′(G) = ∆(G) = r
4 (r ≥ 3), then Σ′(G) ≤ 2nr(2r−1)+n(r−1)(r2+2r−2) , where Σ′(G) is the
1 j 4r k
: edge-chromatic sum of G.
v
i
X
r 1 Introduction
a
In this note we consider graphs which are finite, undirected, and have no loops
or multiple edges. Let V(G) and E(G) denote the sets of vertices and edges of
a graph G, respectively. The degree of a vertex v ∈ V(G) is denoted by dG(v)
′
and the chromatic index of G by χ (G). For a graph G, let ∆(G) and δ(G)
denote the maximum and minimum degrees of vertices in G, respectively. An
(a,b)-biregularbipartitegraphGis abipartitegraphGwiththe verticesinone
part all having degree a and the vertices in the other part all having degree b.
The terms and concepts that we do not define can be found in [5].
A proper edge-coloring of a graph G is a mapping α:E(G)→N such that
′ ′
α(e) 6= α(e) for every pair of adjacent edges e,e ∈ E(G). If α is a proper
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′
edge-coloring of a graph G, then Σ(G,α) denotes the sum of the colors of the
′
edges of G. For a graph G, define the edge-chromatic sum Σ(G) as follows:
′ ′
Σ(G) = minαΣ(G,α), where minimum is taken among all possible proper
edge-colorings of G. A proper t-coloring is a proper edge-coloring which makes
use of t different colors. If α is a proper t-coloring of G and v ∈ V(G), then
S(v,α)denotessetofcolorsappearingonedgesincidenttov. LetGbeagraph
and R ⊆ V(G). A proper edge-coloring of a graph G with colors 1,...,t is
called an R-sequential t-coloring if the edges incident to each vertex v ∈R are
colored by the colors 1,...,dG(v).
The concept of sequential edge-coloring of graphs was introduced by Asra-
tian [1]. In [1, 2], he proved the following result.
Theorem1. IfG=(X∪Y,E)isabipartitegraphwithdG(x)≥dG(y)forevery
xy ∈E(G),wherex∈X andy ∈Y,thenGhasanX-sequential∆(G)-coloring.
On the other hand, in [2] Asratian and Kamalian showed that the problem
of deciding whether a bipartite graph G = (X ∪Y,E) with ∆(G) = 3 has an
X-sequential3-coloringisNP-complete. Someotherresultsonsequentialedge-
coloringsofgraphswereobtainedin[3,4]. Inparticular,in[4]Kamalianproved
the following result.
Theorem 2. If G is an (r −1,r)-biregular (r ≥ 3) bipartite graph with n
rn
vertices, then G has an R-sequential r-coloring with |R|≥ .
l2r−1m
In this note we generalize last theorem. As a corollary, we also obtain the
′
following result: if G is a graph with ∆(G)−δ(G) ≤ 1 and χ(G) =∆(G) =r
(r ≥3), then Σ′(G)≤ 2nr(2r−1)+n(r−1)(r2+2r−2) .
j 4r k
2 The Result
Theorem 3. If G a graph with ∆(G)−δ(G) ≤ 1 and χ′(G) = ∆(G) = r
(r ≥ 3), then G has an R-sequential r-coloring with |R| ≥ (r−1)nr+n , where
l r m
n=|V(G)| and nr =|{v∈V(G):dG(v)=r}|.
Proof. Sinceχ′(G)=∆(G)=r,thereexistsaproperr-coloringαofthegraph
G with colors 1,2,...,r. For i=1,2,...,r, define the set Vα(i) as follows:
Vα(i)={v ∈V(G):i∈/ S(v,α)}.
′ ′′ ′ ′′
Clearly, for any i,i ,1≤i <i ≤r, we have
r
′ ′′
Vα(i)∩Vα(i )=∅ and Vα(i)=V(G)\Vr,
iS=1
2
where Vr ={v∈V(G):dG(v)=r}.
Hence,
r r
n−nr =|V(G)|−|Vr|=(cid:12) Vα(i)(cid:12)= |Vα(i)|.
(cid:12)(cid:12)iS=1 (cid:12)(cid:12) iP=1
(cid:12) (cid:12)
This implies that there exists i0, 1 ≤ i0 ≤ r, for which |Vα(i0)| ≥ n−rnr .
Let R=Vr ∪Vα(i0). Clearly, |R|≥nr+ n−rnr . (cid:6) (cid:7)
(cid:6) (cid:7)
If i = r, then α is an R-sequential r-coloring of G; otherwise define an
0
edge-coloring β as follows: for any e∈E(G), let
α(e), if α(e)6=i ,r,
0

β(e)= i , if α(e)=r,
 0
r, if α(e)=i .
0

It is easy to see that β is an R-sequential r-coloring of G with |R| ≥
(r−1)nr+n . (cid:3)
l r m
Corollary 1. If G is an (r −1,r)-biregular (r ≥ 3) bipartite graph with n
rn
vertices, then G has an R-sequential r-coloring with |R|≥ .
l2r−1m
Corollary 2. If G is a graph with ∆(G)−δ(G) ≤ 1 and χ′(G) = ∆(G) = r
(r ≥3), then
Σ′(G)≤ 2nr(2r−1)+n(r−1)(r2+2r−2)
j 4r k
Proof. Let α be an R-sequential r-coloring of G with |R| ≥ (r−1)nr+n
l r m
described in the proof of Theorem 3. Now, we have
Σ′(G)≤Σ′(G,α) ≤ nr·r(2r+1) +(cid:6)n−rnr(cid:7)r(r2−1) +(cid:0)n−nr−(cid:6)n−rnr(cid:7)(cid:1)(r+2)2(r−1)
2
≤ nr·r(2r+1) + (n−nr2)rr(r−1) +(cid:0)n−nr− n−rnr(cid:1)(r+2)2(r−1)
2
nr·r(r+1) + (n−nr)r(r−1) + (n−nr)(r+2)(r−1)2
= 2 2r 2r
2
nr·r(r+1) (n−nr)(r−1)(r2+2r−2)
= +
4 4r
2nr(2r−1)+n(r−1)(r2+2r−2)
= .
4r
(cid:3)
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References
[1] A.S. Asratian, Investigationofsome mathematicalmodel of scheduling the-
ory, Doctoral Thesis, Moscow, 1980.
[2] A.S. Asratian, R.R. Kamalian, Interval colorings of edges of a multigraph,
Appl. Math. 5 (1987) 25-34 (in Russian).
[3] R.R. Kamalian, Interval edge-colorings of graphs, Doctoral Thesis, Novosi-
birsk, 1990.
[4] R.R.Kamalian,Onanumberofverticeswithanintervalspectruminproper
edgecoloringsofsomegraphs,LiTH-MAT-R-2011/03-SE,LinkopingUniver-
sity, 2011.
[5] D.B. West, Introduction to Graph Theory,Prentice-Hall,New Jersey,2001.
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