Growth rates of the bipartite Erd\H{o}s-Gy\'{a}rf\'{a}s function

TL;DR

This paper investigates the asymptotic growth of the bipartite Erdős-Gyárfás function for various bipartite graphs, providing new bounds and extending existing methods to analyze edge-colorings that ensure multiple colors in each copy of a subgraph.

Contribution

It extends the study of the bipartite Erdős-Gyárfás function to unbalanced complete bipartite graphs, offering new bounds and applying advanced coloring techniques.

Findings

Established thresholds for growth rates of the function.

Derived lower bounds for the balanced bipartite case.

Improved previous bounds by Axenovich, Füredi, and Mubayi.

Abstract

Given two graphs $G, H$ and a positive integer $q$, an $(H,q)$-coloring of $G$ is an edge-coloring of $G$ such that every copy of $H$ in $G$ receives at least $q$ distinct colors. The bipartite Erd\H{o}s-Gy\'{a}rf\'{a}s function $r(K_{n,n}, K_{s,t}, q)$ is defined to be the minimum number of colors needed for $K_{n,n}$ to have a $(K_{s,t}, q)$-coloring. For balanced complete bipartite graphs $K_{p,p}$, the function $r(K_{n,n}, K_{p,p}, q)$ was studied systematically in [Axenovich, F\"{u}redi and Mubayi, {\it J. Combin. Theory Ser. B} {\bf 79} (2000), 66--86]. In this paper, we study the asymptotic behavior of this function for complete bipartite graphs $K_{s,t}$ that are not necessarily balanced. Our main results deal with thresholds and lower and upper bounds for the growth rate of this function, in particular for (sub)linear and (sub)quadratic growth. We also obtain new lower bounds for the balanced bipartite case, and improve several results given by Axenovich, F\"{u}redi and Mubayi. Our proof techniques are based on an extension to bipartite graphs of the recently developed Color Energy Method by Pohoata and Sheffer and its refinements, and a generalization of an old result due to Corr\'{a}di.