A note on the Alon-Saks-Seymour problem

2026-05-27

AI Summary

This work addresses the problem of determining the maximum chromatic number $ f(k) $ of graphs whose edge set can be partitioned into at most $ k $ complete bipartite subgraphs. By employing combinatorial analysis and recursive inequality techniques, the authors establish a key recurrence relation $ f(k+1) \leq f(k) + f(\lfloor k/4 \rfloor) $, which yields a concise and tight exponential upper bound $ f(k) \leq 2^{(\log_2(4k))^2/4} $. This result asymptotically improves the exponent in the best-known upper bound by Mubayi and Vishwanathan by nearly a factor of two and closely approaches the current lower bound up to lower-order terms in the exponent, thereby significantly advancing the understanding of this classical problem.

Abstract

Let $f(k)$ be the maximum possible chromatic number of a graph whose edge set can be partitioned into at most $k$ complete bipartite graphs. Alon, Saks, and Seymour conjectured that $f(k)=k+1$ for all $k$. While the conjecture was verified for $k \leq 9$ by Gao et al., it was disproved by Huang and Sudakov, and further Balodis et al. proved that $f(k) \geq 2^{\widetildeΩ((\log k)^2)}$. In this note, we give a simple proof of the recursive upper bound $f(k+1) \leq f(k)+f(\lfloor k/4 \rfloor)$. Consequently, $f(k) \leq 2^{(\log_2 (4k))^2/4}$ for $k \geq 1$. This improves the previous best known upper bound of Mubayi and Vishwanathan in the exponent by a factor which is asymptotically two. Note that these bounds are sharp up to a lower order factor in the exponent by the result of Balodis et al.

Problem

• chromatic number
• complete bipartite graphs
• Alon-Saks-Seymour conjecture
• graph edge partition
• combinatorics

Innovation

• Alon-Saks-Seymour conjecture
• chromatic number
• complete bipartite partition
• recursive upper bound
• graph coloring