Elsevier

Discrete Applied Mathematics

Volume 175, 1 October 2014, Pages 129-134
Discrete Applied Mathematics

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On r-equitable chromatic threshold of Kronecker products of complete graphs

https://doi.org/10.1016/j.dam.2014.05.036Get rights and content
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Abstract

Let r and k be positive integers. A graph G is r-equitably k-colorable if its vertex set can be partitioned into k independent sets, any two of which differ in size by at most r. The r-equitable chromatic threshold of a graph G, denoted by χr=(G), is the minimum k such that G is r-equitably k-colorable for all kk. Let G×H denote the Kronecker product of graphs G and H. In this paper, we completely determine the exact value of χr=(Km×Kn) for general m,n and r. As a consequence, we show that for r2, if n1r1(m+r)(m+2r1) then Km×Kn and its spanning supergraph Km(n) have the same r-equitable colorability, and in particular χr=(Km×Kn)=χr=(Km(n)), where Km(n) is the complete m-partite graph with n vertices in each part.

Keywords

Equitable coloring
r-Equitable coloring
r-Equitable chromatic threshold
Kronecker product
Complete graph

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Supported by the National Natural Science Foundation of China  (No. 11301410) and the Specialized Research Fund for the Doctoral Program of Higher Education (No. 20130203120021).