Preorder induced by rainbow forbidden subgraphs
Shun-ichi Maezawa, Akira Saito
TL;DR
This work investigates rainbow forbidden subgraphs in edge-colored complete graphs through a preorder $\le$ on connected rainbow-free graphs, focusing on pairs where neither graph is a subgraph of the other. It develops invariants that reflect $\le$, and then analyzes explicit cycle- and tree-related cases to reveal a rich landscape of comparable pairs beyond simple subgraph containment. Key findings include cycle-vs-cycle relations such as $Z_1\le C_k$ for $k\ge4$ and $C_k\le C_{t(k-2)+2}$, and numerous tree-vs-cycle and tree-vs-tree results (e.g., $S_{2,2,1}\le C_k$ for $k\ge4$, $S_{3,1,1}\le C_k$ for $k\ge5$, and $B\le C_k$ for large $k$). The paper also structures the induced poset $(\mathcal H/\equiv,\le)$, identifying the minimum element $\{K_{1,3},K^+_{1,3}\}$ and initial successors, and discusses the extent of known equivalence classes, raising open problems about further nontrivial classes and the poset's global shape.
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 $\le$.