Local Fan-type obstructions and Ore-type conditions for rainbow Hamiltonicity in graph systems
Year: 2026
Type: preprint
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 $\sum_{v\in V}δ(v)\ge\lceil n^2/2\rceil$ guarantees a rainbow Hamilton cycle. We further prove that the local Ore-type condition \[ δ(u)+δ(v)
Source: arXiv (Cornell University)
Authors: Chen, Ailian, Chen, Hao
Institution: Fuzhou University
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7221757540