Sigma Coloring on Powers of Paths and Some Families of Snarks

Abstract

Consider a vertex coloring of a graph where each color is represented by a natural number. The color sum of a vertex is the sum of the colors of its adjacent vertices. The Sigma Coloring Problem concerns determining the sigma chromatic number of a graph G, σ(G), which is the least number of colors for a coloring of G such that the color sum of any two adjacent vertices are different. In this article, we proved that σ(Pnk)≤3 when 2≤k≤n3−1, and we determined the sigma chromatic number for Pnk in the remaining cases. This article also presents the sigma chromatic number for Blanuša 1st and 2nd families of snarks, Flower snarks, Goldberg and Twisted Goldberg snarks.