Dhurandhar, M (1982) Improvement On Brooks Chromatic Bound For A Class Of Graphs. In: Discrete Mathematics, 42 (1). pp. 51-56.
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Brooks' Theorem says that if for a graph G,Δ(G)=n, then G is n-colourable, unless (1) n=2 and G has an odd cycle as a component, or (2) n>2 and Kn+1 is a component of G. In this paper we prove that if a graph G has none of some three graphs (K1,3;K5−e and H) as an induced subgraph and if Δ(G)greater-or-equal, slanted6 and d(G)<Δ(G), then χ(G)<Δ(G). Also we give examples to show that the hypothesis Δ(G)greater-or-equal, slanted6 can not be non-trivially relaxed and the graph K5−e can not be removed from the hypothesis. Moreover, for a graph G with none of K1,3;K5−e and H as an induced subgraph, we verify Borodin and Kostochka's conjecture that if for a graph G,Δ(G)greater-or-equal, slanted9 and d(G)<Δ(G), then χ(G)<Δ(G).
|Item Type:||Journal Article|
|Additional Information:||Copyright of this article belongs to Elsevier Science.|
|Department/Centre:||Division of Physical & Mathematical Sciences > Mathematics|
|Date Deposited:||20 Aug 2009 11:47|
|Last Modified:||14 Jun 2011 09:04|
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