Pancyclicity in Graph Families with the Ore-Type Condition

TL;DR

This paper proves that under certain Ore-type degree sum conditions, a family of graphs on the same vertex set contains rainbow cycles of all lengths, supporting the rainbow pancyclicity conjecture and establishing sharp results.

Contribution

It extends previous rainbow Hamiltonian cycle results to rainbow pancyclicity and vertex-pancyclicity under Ore-type conditions, with sharp extremal examples.

Findings

Either each vertex lies in rainbow cycles of all lengths from 4 to n, or all graphs are complete bipartite with equal parts.

The paper establishes rainbow pancyclicity of the graph family under the Ore-type condition.

Provides extremal graph examples showing the results are sharp.

Abstract

Let $ n \in \mathbb{N} $ with $ n \geq 3 $, and let $\mathcal{G} = \{G_i:i\in [n]\} $ be a family of $ n $-vertex graphs on a common vertex set $V$, where the graphs in the family do not need to be distinct. A graph $H$ with vertex set $V$ is \emph{rainbow} in $\mathcal{G}$ if there exists an injection $ \phi: E(H) \to [n] $ such that $e \in E(G_{\phi(e)})$ for every edge $e \in E(H)$, where $|E(H)|\leq n$. In 2020, Joos and Kim proved that $\mathcal{G}$ contains a rainbow Hamiltonian cycle under the Dirac-type condition. Recently, Liu, Chen, and Ma generalized this result by replacing the Dirac-type condition with a more general Ore-type condition involving degree sums of non-adjacent vertices: If $\sigma(\mathcal{G}) \geq n$, then $\mathcal{G}$ contains a rainbow Hamiltonian cycle, where the Ore-type condition $\sigma(\mathcal{G})$ is defined as follows: $ \sigma(\mathcal{G}) = \min\{d_p(u) + d_q(v) \mid uv \notin E(G_i) \text{ for some } i \in [n] \text{ and for all } p, q \in [n]\}. $ In this paper, under the Ore-type condition, we show that either each vertex of $V$ is contained in a rainbow cycle of length $\ell$ for every $\ell\in[4,n]$, or $G_1=\cdots=G_n=K_{\frac{n}{2},\frac{n}{2}}$. As a corollary, we deduce the rainbow pancyclicity of $\mathcal{G}$, which supports the famous meta-conjecture posed by Bondy. Furthermore, we prove rainbow vertex-pancyclicity of $\mathcal{G}$ under the Ore-type condition and provide an extremal graph family to show that the result is sharp.