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Efficient Graph Packing via Game Colouring

Published online by Cambridge University Press:  01 September 2009

H. A. KIERSTEAD  and
A. V. KOSTOCHKA
H. A. KIERSTEAD
Affiliation:
Department of Mathematics and Statistics, Arizona State University, Tempe, AZ 85287, USA (e-mail: kierstead@asu.edu)
A. V. KOSTOCHKA
Affiliation:
Department of Mathematics, University of Illinois, Urbana, IL 61801, USA and Institute of Mathematics, Novosibirsk, Russia (e-mail: kostochk@math.uiuc.edu)
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Abstract

The game colouring number gcol(G) of a graph G is the least k such that, if two players take turns choosing the vertices of a graph, then either of them can ensure that every vertex has fewer than k neighbours chosen before it, regardless of what choices the other player makes. Clearly gcol(G) ≤ Δ(G)+1. Sauer and Spencer [20] proved that if two graphs G1 and G2 on n vertices satisfy 2Δ(G1)Δ(G2) < n then they pack, i.e., there is an embedding of G1 into the complement of G2. We improve this by showing that if (gcol(G1)−1)Δ(G2)+(gcol(G2)−1)Δ(G1) < n then G1 and G2 pack. To our knowledge this is the first application of colouring games to a non-game problem.

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 18 , Issue 5: New Directions in Algorithms, Combinatorics and Optimization , September 2009 , pp. 765 - 774
Copyright
Copyright © Cambridge University Press 2009

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