A counterexample to the Alon-Saks-Seymour conjecture and related problems

Authors:
Hao Huang;Benny Sudakov
Affiliations:
Department of Mathematics UCLA, 90095, Los Angeles, CA, USA;Department of Mathematics UCLA, 90095, Los Angeles, CA, USA
Venue:
Combinatorica
Year:
2012

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Note: Some improved bounds on communication complexity via new decomposition of cliques

Discrete Applied Mathematics

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Abstract

Consider a graph obtained by taking an edge disjoint union of k complete bipartite graphs. Alon, Saks, and Seymour conjectured that such graphs have chromatic number at most k+1. This well known conjecture remained open for almost twenty years. In this paper, we construct a counterexample to this conjecture and discuss several related problems in combinatorial geometry and communication complexity.