Average distance in colored graphs
Peter Dankelmann, Wayne Goddard, Peter Raymond Slater · Journal of Graph Theory · 2001
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