Advertisement

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 {(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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Dilworth, R.P.: A decomposition theorem for partially ordered sets. Annals of Mathematics 51, 161–166 (1950)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Gerke, S., McDiarmid, C.: Graph Imperfection. Journal of Combinatorial Theory B 83, 58–78 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Gerke, S., McDiarmid, C.: Graph Imperfection II. Journal of Combinatorial Theory B 83, 79–101 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Halldórsson, M.M., Kortsarz, G.: Tools for Multicoloring with Applications to Planar Graphs and Partial k-Trees. Journal of Algorithms 42, 334–366 (2002)zbMATHCrossRefGoogle Scholar
  5. 5.
    Hochbaum, D.S.: Efficient bounds for the stable set, vertex cover and set packing problems. Discrete Applied Mathematics 6, 243–254 (1983)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    McDiarmid, C., Reed, B.: Channel Assignment and Weighted Coloring. Networks 36, 114–117 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    McDiarmid, C.: Discrete Mathematics and Radio Channel Assignment. Appears in Recent Advances in Algorithms and Combinatorics. Springer, Heidelberg (2003)Google Scholar
  8. 8.
    Miyamoto, Y., Matsui, T.: Linear Time Approximation Algorithm for Multicoloring Lattice Graphs with Diagonals. Journal of the Operations Research Society of Japan 47, 123–128 (2004)zbMATHMathSciNetGoogle Scholar
  9. 9.
    Miyamoto, Y., Matsui, T.: Multicoloring Unit Disk Graphs on Triangular Lattice Points. In: Proceedings of the Sixteenth ACM-SIAM Symposium on Discrete Algorithms (to appear)Google Scholar
  10. 10.
    Narayanan, L., Shende, S.M.: Static Frequency Assignment in Cellular Networks. Algorithmica 29, 396–409 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Schrijver, A.: Combinatorial Optimization. Springer, Heidelberg (2003)zbMATHGoogle Scholar

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

Personalised recommendations