Ore's Theorem for rainbow Hamiltonian-connected graphs

TL;DR

This paper extends Ore's Theorem to rainbow Hamiltonian paths and cycles in collections of graphs, establishing conditions under which such rainbow structures exist, and also allows embedding prescribed rainbow linear forests.

Contribution

It introduces a new Ore-type condition for rainbow Hamiltonian connectivity and cycles in graph collections, including embedding of rainbow linear forests.

Findings

Rainbow Hamiltonian paths exist between every pair of vertices under the given degree condition.

Either a rainbow Hamiltonian cycle exists or all pairs are connected by rainbow Hamiltonian paths.

The theorem is strengthened to include embedding prescribed rainbow linear forests.

Abstract

Let $G = (G_1, G_2, \ldots, G_m)$ be a collection of $m$ graphs on a common vertex set $V$. For a graph $H$ with vertices in $V$, we say that $G$ contains a rainbow $H$ if there is an injection $c: E(H) \to [m]$ such that for every edge $e \in E(H)$, we have $e \in E(G_{c(e)})$. In this paper, we show that if $G = (G_1, \ldots, G_n)$ is a collection of graphs on $n$ vertices such that for every $i \in [n]$, $d_{G_i}(u) + d_{G_i}(v) \geq n$ whenever $uv \notin E(G_i)$, then either $G$ contains rainbow Hamiltonian paths between every pair of vertices, or $G$ contains a rainbow Hamiltonian cycle. Moreover, we prove a stronger version in which we may also embed prescribed rainbow linear forests into the Hamiltonian paths.