Geodesic
A new upper bound for the chromatic number of a graph
Discussiones Mathematicae. Graph Theory, Tome 27 (2007) no. 1, pp. 137-142

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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 K_ω+3- C₅.
Keywords: Vertex colouring, chromatic number, upper bound 
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Schiermeyer, Ingo. A new upper bound for the chromatic number of a graph. Discussiones Mathematicae. Graph Theory, Tome 27 (2007) no. 1, pp. 137-142. http://geodesic.mathdoc.fr/item/DMGT_2007_27_1_a11/