Perfectness and Imperfectness of the kth Power of Lattice Graphs

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 e1 def. = (1, 0), e2 def. = (1/2, √ 3/2), and an edge set consists of a pair of vertices with unit distance. Let S(m,n) and T (m,n) be the kth power of the graph S(m,n) and T (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(m,n) is perfect] and/or [∀m, T (m,n) is perfect], respectively. These conditions imply polynomial time approximation algorithms for multicoloring (S(m,n), w) and (T (m,n), w).