Rainbow Hamiltonian paths in graph collections under edge-size conditions

Abstract

Let $\mathbf{G}=\{G_1,\dots,G_{n-1}\}$ be a collection of not necessarily distinct graphs of order $n$ on the same vertex set $V$. A path $P$ with $V(P)\subseteq V$ is called \emph{rainbow} in $\mathbf{G}$, if there exists an injection $ϕ\colon E(P)\to [n-1]$ such that $e\in E(G_{ϕ(e)})$ for each $e\in E(P)$. In this paper, we show that if $n\ge3$, $e(G_i)> \binom{n-2}{2}+2$ for each $i\in[n-1]$ and $G_j$ is connected for some $j\in[n-1]$, then there exists a rainbow Hamiltonian path in $\mathbf{G}$ unless $n=6$ with $G_1=G_2=\dots=G_{5}\cong K_2\vee 4K_1$, or $n=8$ with $G_1=G_2=\dots=G_{7}\cong K_3\vee 5K_1$. This corrects Theorem 6 in [Graphs Combin., \textbf{41}(6)(2025), 119] and relaxes its hypothesis from requiring all $G_i$ to be connected to only requiring that at least one $G_i$ is connected, where $i\in[n-1]$. Moreover, we show that if $n\ge2$ and $e(G_i)\ge \binom{n-1}{2}$ for each $i\in[n-1]$, then $\mathbf{G}$ contains a rainbow Hamiltonian path unless $G_1=G_2=\dots=G_{n-1}\cong K_{n-1}\cup K_1$, or $n=4$ with $G_1=G_2=G_3\cong K_{1,3}$. As a consequence, Theorem 2 of [Discrete Math., \textbf{347}(2024), 114128] follows immediately.