Preorder induced by rainbow forbidden subgraphs

TL;DR

This paper explores the preorder relation between graphs based on rainbow subgraph containment in edge-colored complete graphs, focusing on pairs of graphs that are not subgraphs of each other and characterizing their structure.

Contribution

It extends previous work by analyzing pairs of graphs with no subgraph relation under the rainbow containment preorder, revealing many such pairs and studying their properties.

Findings

Many such pairs $(H_1, H_2)$ exist with $H_1 subseteq H_2$ and $H_2 subseteq H_1$.

The structure of these pairs under the rainbow containment preorder is characterized.

The paper generalizes the understanding of rainbow subgraph relations beyond subgraph inclusion.

Abstract

A subgraph $H$ of an edge-colored graph $G$ is rainbow if all the edges of $H$ receive different colors. If $G$ does not contain a rainbow subgraph isomorphic to $H$, we say that $G$ is rainbow $H$-free. For connected graphs $H_1$ and $H_2$, if every rainbow $H_1$-free edge-colored complete graph colored in sufficiently many colors is rainbow $H_2$-free, we write $H_1\le H_2$. The binary relation $\le$ is reflexive and transitive, and hence it is a preorder. If $H_1$ is a subgraph of $H_2$, then trivially $H_1\le H_2$ holds. On the other hand, there exists a pair $(H_1, H_2)$ such that $H_1$ is a proper supergraph of $H_2$ and $H_1\le H_2$ holds. Cui et al.~[Discrete Math.~\textbf{344} (2021) Article Number 112267] characterized these pairs. In this paper, we investigate the pairs $(H_1, H_2)$ with $H_1\le H_2$ when neither $H_1$ nor $H_2$ is a subgraph of the other. We prove that there are many such pairs and investigate their structure with respect to $\le$.