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An upper bound for the total chromatic number of dense graphs
Author(s) -
Hind H. R.
Publication year - 1992
Publication title -
journal of graph theory
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 54
eISSN - 1097-0118
pISSN - 0364-9024
DOI - 10.1002/jgt.3190160302
Subject(s) - combinatorics , mathematics , upper and lower bounds , vertex (graph theory) , degree (music) , graph , chromatic scale , discrete mathematics , order (exchange) , integer (computer science) , computer science , physics , mathematical analysis , finance , acoustics , economics , programming language
In this paper we consider those graphs that have maximum degree at least 1/ k times their order, where k is a (small) positive integer. A result of Hajnal and Szemerédi concerning equitable vertex‐colorings and an adaptation of the standard proof of Vizing's Theorem are used to show that if the maximum degree of a graph G satisfies Δ( G ) ≥ | V ( G )/ k , then X ″( G ) ≤ Δ( G ) + 2 k + 1. This upper bound is an improvement on the currently available upper bounds for dense graphs having large order.