Almost-rainbow edge-colorings of some small subgraphs

TL;DR

This paper improves bounds on the minimum number of colors needed to edge-color complete graphs and bipartite graphs so that small subgraphs have a certain diversity of colors, advancing understanding in graph coloring problems.

Contribution

The paper provides new bounds and constructions for edge-colorings of complete and bipartite graphs ensuring diverse coloring of small subgraphs, improving previous results.

Findings

Improved bounds on f(n,p,q) for complete graphs.

Constructed n-color edge-coloring of K_{n,n} with 3 colors on every C_4.

Enhanced previous bounds by Axenovich, Erdős, and Gyárfás.

Abstract

Let $f(n,p,q)$ be the minimum number of colors necessary to color the edges of $K_n$ so that every $K_p$ is at least $q$-colored. We improve current bounds on the {7/4}n-3$, slightly improving the bound of Axenovich. We make small improvements on bounds of Erd\H os and Gy\'arf\'as by showing ${5/6}n+1\leq f(n,4,5)$ and for all even $n\not\equiv 1 \pmod 3$, $f(n,4,5)\leq n-1$ . For a complete bipartite graph $G=K_{n,n}$, we show an n-color construction to color the edges of $G$ so that every $C_4\subseteq G$ is colored by at least three colors. This improves the best known upper bound of M. Axenovich, Z. F\"uredi, and D. Mubayi.