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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## Abstract

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 |
---|---|

Publication: | Discrete Mathematics |

Publisher: | Elsevier Science |

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 |

URI: | http://eprints.iisc.ac.in/id/eprint/22185 |

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