Article

# Colourings of the Cartesian Product of Graphs and Multiplicative Sidon Sets

Department of Mathematics, Case Western Reserve University, Cleveland, U.S.A.; Departament de Matemàtica Aplicada II, Universitat Politècnica de Catalunya, Barcelona, Spain
Electronic Notes in Discrete Mathematics DOI:10.1016/j.endm.2007.01.006 pp.33-40
Source: arXiv

ABSTRACT Let F be a family of connected bipartite graphs, each with at least three vertices. A proper vertex colouring of a graph G with no bichromatic subgraph in F is F-free. The F-free chromatic numberχ(G,F) of a graph G is the minimum number of colours in an F-free colouring of G. For appropriate choices of F, several well-known types of colourings fit into this framework, including acyclic colourings, star colourings, and distance-2 colourings. This paper studies F-free colourings of the cartesian product of graphs. Let H be the cartesian product of the graphs G1,G2,…,Gd. Our main result establishes an upper bound on the F-free chromatic number of H in terms of the maximum F-free chromatic number of the Gi and the following number-theoretic concept. A set S of natural numbers is k-multiplicative Sidon if ax=by implies a=b and x=y whenever x,y∈S and 1⩽a,b⩽k. Suppose that χ(Gi,F)⩽k and S is a k-multiplicative Sidon set of cardinality d. We prove that χ(H,F)⩽1+2k⋅maxS. We then prove that the maximum density of a k-multiplicative Sidon set is Θ(1/logk). It follows that χ(H,F)⩽O(dklogk).

0 0
·
0 Bookmarks
·
43 Views
• ##### Conference Proceeding:Powers of Geometric Intersection Graphs and Dispersion Algorithms.
Algorithm Theory - SWAT 2002, 8th Scandinavian Workshop on Algorithm Theory, Turku, Finland, July 3-5, 2002 Proceedings; 01/2002
• ##### Article:On Powers of Chordal Graphs And Their Colorings
[show abstract] [hide abstract]
ABSTRACT: The k-th power of a graph G is a graph on the same vertex set as G, where a pair of vertices is connected by an edge if they are of distance at most k in G. We study the structure of powers of chordal graphs and the complexity of coloring them. We start by giving new and constructive proofs of the known facts that any power of an interval graph is an interval graph, and that any odd power of a general chordal graph is again chordal. We then show that it is computationally hard to approximately color the even powers of n- vertex chordal graphs within an n 1 2 Gammaffl factor, for any ffl ? 0. We present two exact and closed formulas for the chromatic polynomial for the k-th power of a tree on n vertices. Furthermore, we give an O(kn) algorithm for evaluating the polynomial. Keywords: Chordal graphs, chromatic number, chromatic polynomial, coloring, interval graphs, power of a graph, tree. 1 Introduction In this paper we study the structure of powers of chordal graphs a...
12/2000;
• ##### Conference Proceeding:Coloring powers of planar graphs.
Proceedings of the Eleventh Annual ACM-SIAM Symposium on Discrete Algorithms, January 9-11, 2000, San Francisco, CA, USA.; 01/2000

Available from
5 Feb 2013

### Keywords

appropriate choices

bichromatic subgraph

bipartite graphs

cardinality d

cartesian product

colours

distance-2 colourings

F-free chromatic number

following number-theoretic concept

graph G

graphs

k-multiplicative Sidon

main result

maximum density

maximum F-free chromatic number

minimum number

paper studies F-free colourings

proper vertex colouring