Average distance in colored graphs

  • Authors:

  • Peter Dankelmann;Wayne Goddard;Peter Slater

  • Affiliations:

  • School of Mathematical and Statistical Sciences, University of Natal, Durban, 4041, South Africa;School of Geological and Computer Sciences, University of Natal, Durban, 4041, South Africa;Department of Mathematics and Statistics, University of Alabama in Huntsville, Huntsville Alabama 85899

  • Venue:
  • Journal of Graph Theory
  • Year:
  • 2001

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Abstract

For a graph G where the vertices are colored, the colored distance of G is defined as the sum of the distances between all unordered pairs of vertices having different colors. Then for a fixed supply s of colors, ds(G) is defined as the minimum colored distance over all colorings with s. This generalizes the concepts of median and average distance. In this paper we explore bounds on this parameter, especially a natural lower bound and the particular case of balanced 2-colorings (equal numbers of red and blue). We show that the general problem is NP-hard but there is a polynomial-time algorithm for trees. © 2001 John Wiley & Sons, Inc. J Graph Theory 38: 1–17, 2001