Tree versus tree of preorder induced by rainbow forbidden subgraphs

TL;DR

The paper studies rainbow forbidden subgraphs on edge-colored complete graphs by examining the preorder $H_1 \le H_2$ and the induced equivalence relation $\equiv$, focusing on trees within a restricted class of connected graphs. It establishes a near-complete classification: for trees of different order, the only nontrivial equivalence class is the pair $\{K_{1,k},K_{1,k}^+\}$ with $k\ge3$, and if $|V(T_1)|>|V(T_2)|\ge4$ and $T_1\le T_2$, then $(T_1,T_2)=(K_{1,k}^+,K_{1,k})$. For trees of the same order, the authors prove there are no equivalence classes when degrees differ, and they list all comparable pairs with $ds(T_1)\neq ds(T_2)$, including $(K_{1,k}^+,K_{1,k+1})$, $(F'_{k-1},F_{k-1})$, and $(B^*_{k,k-2},B^*_{k-1,k-1})$, plus a few small cases. The work uses rainbow colorings, structural lemmas, and explicitly defined tree families (e.g., $F_m$, $F_m'$, $B^*_{s,t}$, and $S_{a,b,c}$) to derive necessary conditions and complete the characterization. It also highlights unresolved instances and poses open questions, contributing to the broader understanding of rainbow extremal phenomena in graph theory.

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 there exists an integer $t=t(H_1,H_2)$ such that every rainbow $H_1$-free edge-colored complete graph colored with $t$ or more colors is rainbow $H_2$-free, then we write $H_1\le H_2$. The binary relation $\le$ is reflexive and transitive, and hence it is a preorder. For graphs $H_1$ and $H_2$, we write $H_1 \equiv H_2$ if both $H_1 \le H_2$ and $H_2 \le H_1$ hold. Then $\equiv$ is an equivalence relation. 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. Q.~Cui, Q.~Liu, C.~Magnant and A.~Saito [Discrete Math. {\bf 344} (2021) Article Number 112267] characterized these pairs. %On the other hand, there are few known results regarding the study of $\leq$ for the incomparable with respect to $\subseteq$. Cui et al. found pairs of graphs $H_1$ and $H_2$ such that $H_1 \leq H_2$ and $H_2 \leq H_1$, that is, non-singleton equivalence class with respect to $\le$. However, we have not found any other non-singleton equivalence class with respect to $\le$ except for those discovered by Cui et al. In this paper. we investigate the existence of non-singleton equivalence class with respect to $\le$ by focusing on trees.

Figures (15)

• Figure 1: The graph $G$ consists of the top vertex and several triangles and $G^{\prime}$ is obtained from $G$ by deleting $e_1$ and by adding $e^{\prime}$.
• Figure 2: The left figure represents a graph $F_m$ and the right one represents a graph $F_m^{\prime}$.
• Figure 3: The figure represents a graph $B^*_{s,t}$.
• Figure 4: The dotted lines represent deleted edges and the bold lines represent added edges. The resulting graphs are isomorphic to $K_{1,3}^+$.
• Figure 5: The dotted lines represent deleted edges and the bold lines represent added edges. The resulting graphs are isomorphic to $S_{1,2,3}$.

Theorems & Definitions (24)

• Theorem A: Cui, Liu, Magnant, and Saito CLMS
• Theorem B: Maezawa and Saito maezawa
• Theorem 1
• Theorem 2
• Theorem 3
• Lemma 4
• Theorem 5
• Claim 3.1
• Claim 3.2
• Claim 3.3