Spectral radius and rainbow Hamiltonicity in bipartite graphs

TL;DR

This paper establishes spectral radius conditions that guarantee the existence of rainbow Hamilton paths and cycles in bipartite graph families, providing tight bounds and characterizations of extremal cases.

Contribution

It introduces spectral radius criteria for rainbow Hamiltonicity in bipartite graphs and characterizes extremal graphs achieving these bounds.

Findings

Spectral radius conditions ensure rainbow Hamilton paths and cycles.

Complete characterization of extremal spectral graphs.

Tight sufficient conditions derived using bi-shifting technique.

Abstract

Let $\mathcal{G}=\{G_1, G_2, \ldots , G_k\}$ be a family of bipartite graphs on the same vertex set. A rainbow Hamilton path (cycle) in $\mathcal{G}$ is a path (cycle) that visits each vertex precisely once such that any two edges belong to different graphs of $\mathcal{G}.$ In this paper, by adopting the technique of bi-shifting, we present tight sufficient conditions in terms of the spectral radius for a family $\mathcal{G}$ to admit a rainbow Hamilton path and cycle, respectively. Meanwhile, we completely characterize the corresponding spectral extremal graphs.