Coloring Graphs with Dense Neighborhoods

It is shown that any graph with maximum degree Δ in which the average degree of the induced subgraph on the set of all neighbors of each vertex exceeds is either -colorable or contains a clique on more than vertices. In the case we improve the bound on the average degree to and the bound on the clique number to . As corollaries, we show that every graph satisfies and every graph satisfies .

[1]  Andrew D. King.  Hitting all maximum cliques with a stable set using lopsided independent transversals , 2009, J. Graph Theory.

[2]  Hal A. Kierstead,et al.  On the choosability of complete multipartite graphs with part size three , 2000, Discret. Math..

[3]  W. Mader.  Existenzn-fach zusammenhängender Teilgraphen in Graphen genügend großer Kantendichte , 1972 .

[4]  Penny E. Haxell.  On the Strong Chromatic Number , 2004, Comb. Probab. Comput..

[5]  B. Reed.  Graph Colouring and the Probabilistic Method , 2001 .

[6]  Landon Rabern,et al.  Conjectures equivalent to the Borodin-Kostochka Conjecture that are a priori weaker , 2012 .

[7]  Daniel W. Cranston,et al.  Coloring claw-free graphs with − 1 colors , 2012 .

[8]  B. Reed.  ω, Δ, and χ , 1998 .

[9]  Penny E. Haxell,et al.  On Forming Committees , 2011, Am. Math. Mon..

[10]  Ron Aharoni,et al.  Independent systems of representatives in weighted graphs , 2007, Comb..

[12]  Tibor Szabó,et al.  Odd Independent Transversals are Odd , 2006, Comb. Probab. Comput..

[13]  Gábor Tardos,et al.  Extremal Problems For Transversals In Graphs With Bounded Degree , 2006, Comb..

[14]  Bruce A. Reed,et al.  A Strengthening of Brooks' Theorem , 1999, J. Comb. Theory B.

[15]  Daniel W. Cranston,et al.  Coloring Claw-Free Graphs with Delta-1 Colors , 2013, SIAM J. Discret. Math..

[16]  Benny Sudakov,et al.  List Colouring When The Chromatic Number Is Close To the Order Of The Graph , 2004, Comb..

[17]  Reinhard Diestel,et al.  Graph Theory , 1997 .

[18]  Alexandr V. Kostochka,et al.  On an upper bound of a graph's chromatic number, depending on the graph's degree and density , 1977, J. Comb. Theory B.