Local Fan-type obstructions and Ore-type conditions for rainbow Hamiltonicity in graph systems

Abstract

Let $G=(G_1,...,G_n)$ be a graph system on a common $n$-vertex set $V$. For $v\in V$, set \[ δ(v)=\min_{i\in[n]}d_{G_i}(v), \qquad E_{\cap}=\bigcap_{i=1}^n E(G_i), \] and let $d^c(u,v)$ denote the length of a shortest rainbow $u$--$v$ path. We study Hamiltonicity conditions imposed only on pairs at small rainbow distance. Suppose that every $G_i$ is $2$-connected and that \[ δ(u),δ(v)<\frac n2, \qquad d^c(u,v)\le2 \quad\Longrightarrow\quad uv\in E_{\cap}. \] Our main result shows that any system satisfying this local Fan-type condition but containing no rainbow Hamilton cycle must satisfy \[ E_{\cap}=\varnothing, \qquad \sum_{v\in V}δ(v)\le\frac{n^2-1}{2}. \] Consequently, either $E_{\cap }\ne \varnothing$ or ${\sum }_{v\in V}\delta (v)\ge \lceil n^2/2\rceil$ guarantees a rainbow Hamilton cycle. We further prove that the local Ore-type condition \[ δ(u)+δ(v)<n, \qquad d^c(u,v)\le2 \quad\Longrightarrow\quad uv\in E_{\cap} \] already forces a rainbow Hamilton cycle. Along the way, the Fan-type hypothesis yields a rainbow Hamilton path and a rainbow $(n-1)$-cycle. For diagonal systems, our conditions reduce to the classical Fan and local Ore conditions. The proofs combine rainbow rotation--extension with a double-counting argument in an auxiliary digraph. Keywords: graph system; rainbow Hamilton cycle; rainbow distance; Fan-type condition; Ore-type condition; rotation--extension.