# Perfectness and Imperfectness of the kth Power of Lattice Graphs

• Yuichiro Miyamoto
• Tomomi Matsui
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3521)

## 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 {(xe 1 + ye 2) | x ∈ {0,1,2,...,m − 1}, y ∈ {0,1,2,...,n − 1}} where $$e_1 = (1,0), e_2 = (1/2, \sqrt{3}/2)$$, and an edge set consists of a pair of vertices with unit distance. Let S k (m,n) and T k (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 $$w : V \longrightarrow Z_+$$, a multicoloring of G is an assignment of colors to V such that each vertex vV 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 [∀ mT 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).

## Keywords

Polynomial Time Undirected Graph Polynomial Time Algorithm Channel Assignment Comparability Graph
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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