Perfectness and Imperfectness of the kth Power of Lattice Graphs

  • Yuichiro Miyamoto
  • Tomomi Matsui
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 {(xe1 + ye2) | 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 Sk(m,n) and Tk(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, theSk(m,n) is perfect] and/or [∀ mTk(m,n) is perfect], respectively. These conditions imply polynomial time approximation algorithms for multicoloring (Sk(m,n),w) and (Tk(m,n),w).

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Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Yuichiro Miyamoto
    • 1
  • Tomomi Matsui
    • 2
  1. 1.Sophia UniversityTokyoJapan
  2. 2.University of TokyoTokyoJapan

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