Perfectness and Imperfectness of the kth Power of Lattice Graphs

Abstract. Given a pair of non-negative integers m and n, S(m, n) denotes a square lattice graph with a vertex set {0, 1, 2,...,m − 1} × {0, 1, 2,...,n − 1}, where a pair of two vertices is adjacent if and only if the distance is equal to 1. A triangular lattice graph T (m, n) has a vertex set {(xe1 + ye2) | x ∈{0, 1, 2,...,m − 1}, y∈{0, 1, 2,...,n − 1}} where def. def. e1 = (1, 0), e2 = (1/2, 3/2), and an edge set consists of a pair of vertices with unit distance. Let S k (m, n) andT k (m, n) bethekth power of the graph S(m, n) andT (m, n), respectively. Given an undirected graph G =(V,E) and a non-negative vertex weight function w: V → Z+, a multicoloring of G is an assignment of colors to V such that each vertex v ∈ V admits w(v) colors and every adjacent pair of two vertices does not share a common color. In this paper, we show necessary and sufficient conditions that [∀m, theS k (m, n) is perfect] and/or [∀m, T k (m, n) is perfect], respectively. These conditions imply polynomial time approximation algorithms for multicoloring (S k (m, n),w)and(T k (m, n),w).