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Sequential edge-coloring on the subset of vertices of almost regular graphs 4 1 0 Petros A. Petrosyan 2 n a Department of Informatics and Applied Mathematics, J Yerevan State University, 0025, Armenia 4 Institute for Informatics and Automation Problems, ] National Academy of Sciences, 0014, Armenia O E-mail: pet [email protected] C h. January 7, 2014 t a m Abstract [ LetGbeagraphandR⊆V(G). Aproperedge-coloringofagraphG 1 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 1 ′ 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) 3 References [1] A.S. Asratian, Investigationofsome mathematicalmodel of scheduling theory, 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. 4