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

TL;DR

This paper constructs a counterexample to the long-standing Alon-Saks-Seymour conjecture, showing that graphs formed by union of bipartite graphs can have higher chromatic number than previously believed, and explores related problems.

Contribution

The paper provides the first known counterexample to the Alon-Saks-Seymour conjecture and discusses implications for combinatorial geometry and communication complexity.

Findings

Counterexample disproves the conjecture

Chromatic number can exceed k+1 in such graphs

Implications for related combinatorial problems

Abstract

Consider a graph obtained by taking edge disjoint union of $k$ complete bipartite graphs. Alon, Saks and Seymour conjectured that such graph has 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.