Equitable colorings of Cartesian products of graphs

  • Authors:

  • Wu-Hsiung Lin;Gerard J. Chang

  • Affiliations:

  • Department of Mathematics, National Taiwan University, Taipei 10617, Taiwan;Department of Mathematics, National Taiwan University, Taipei 10617, Taiwan and Taida Institute for Mathematical Sciences, National Taiwan University, Taipei 10617, Taiwan and National Center for ...

  • Venue:
  • Discrete Applied Mathematics
  • Year:
  • 2012

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Abstract

The present paper studies the following variation of vertex coloring on graphs. A graph G is equitably k-colorable if there is a mapping f:V(G)-{1,2,...,k} such that f(x)f(y) for xy@?E(G) and ||f^-^1(i)|-|f^-^1(j)||@?1 for 1@?i,j@?k. The equitable chromatic number of a graph G, denoted by @g"=(G), is the minimum k such that G is equitably k-colorable. The equitable chromatic threshold of a graph G, denoted by @g"=^*(G), is the minimum t such that G is equitably k-colorable for all k=t. Our focus is on the equitable colorability of Cartesian products of graphs. In particular, we give exact values or upper bounds of @g"=(G@?H) and @g"=^*(G@?H) when G and H are cycles, paths, stars, or complete bipartite graphs.