Elsevier

Discrete Mathematics

Volume 312, Issue 6, 28 March 2012, Pages 1273-1281
Discrete Mathematics

Graphs with chromatic number close to maximum degree

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Abstract

Let G be a color-critical graph with χ(G)Δ(G)=2t+15 such that the subgraph of G induced by the vertices of degree 2t+1 has clique number at most t1. We prove that then either t3 and G=K2t+2 or t=2 and G{K6,O5}, where O5 is a special graph with χ(O5)=5 and |O5|=9. This result for t3 improves a case of a theorem by Rabern (2012) [9] and for t=2 answers a question raised by Kierstead and Kostochka (2009) in [6].

Keywords

Graph coloring
Brooks’ theorem
Critical graphs

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