## A NEW UPPER BOUND FOR THE CHROMATIC NUMBER OF A GRAPH

Ingo Schiermeyer

Institut für Diskrete Mathematik und Algebra
09596 Freiberg, Germany
e-mail: schierme@tu-freiberg.de

## Abstract

Let G be a graph of order n with clique number ω(G), chromatic number χ(G) and independence number α(G). We show that χ(G) ≤ [(n+ω+1−α)/2]. Moreover, χ(G) ≤ [(n+ω−α)/2], if either ω+α = n+1 and G is not a split graph or α+ω = n−1 and G contains no induced Kω+3− C5.

Keywords: Vertex colouring, chromatic number, upper bound.

2000 Mathematics Subject Classification: 05C15, 05C69.

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