Edge-coloring a graph $G$ so that every copy of a graph $H$ has an odd color class

TL;DR

This paper investigates the minimum number of colors needed to edge-color complete graphs and bipartite graphs so that every subgraph isomorphic to a given graph has an odd color class, linking combinatorial design with graph coloring.

Contribution

It introduces the function g(G,H) for odd color class edge-colorings and establishes bounds for complete and bipartite graphs, advancing understanding of H-codes and their applications.

Findings

g(K_n,K_5) ≤ n^{o(1)}

g(K_{n,n}, C_4) = n/2 + o(n)

d_{K_5}(n) ≥ 1/n^{o(1)}

Abstract

Recently, Alon introduced the notion of an $H$-code for a graph $H$: a collection of graphs on vertex set $[n]$ is an $H$-code if it contains no two members whose symmetric difference is isomorphic to $H$. Let $D_{H}(n)$ denote the maximum possible cardinality of an $H$-code, and let $d_{H}(n)=D_{H}(n)/2^{n \choose 2}$. Alon observed that a lower bound on $d_{H}(n)$ can be obtained by attaining an upper bound on the number of colors needed to edge-color $K_n$ so that every copy of $H$ has an odd color class. Motivated by this observation, we define $g(G,H)$ to be the minimum number of colors needed to edge-color a graph $G$ so that every copy of $H$ has an odd color class. We prove $g(K_n,K_5) \le n^{o(1)}$ and $g(K_{n,n}, C_4)= n/2+o(n)$. The first result shows $d_{K_5}(n) \ge \frac{1}{n^{o(1)}}$ and was obtained independently in arXiv:2306.14682.