A spectral condition for Hamilton cycles in tough bipartite graphs

TL;DR

This paper establishes a precise spectral radius condition for balanced bipartite graphs with a certain toughness to contain Hamilton cycles, advancing understanding of spectral graph theory and Hamiltonicity.

Contribution

It provides a sharp spectral radius criterion for Hamiltonicity in balanced bipartite graphs with bipartite toughness at least one, addressing a previously open problem.

Findings

Spectral radius condition guarantees Hamilton cycles in balanced bipartite graphs.

The condition is sharp and improves previous bounds.

Addresses a problem proposed in earlier research.

Abstract

Let $G$ be a graph. The {\em spectral radius} of $G$ is the largest eigenvalue of its adjacency matrix. For a non-complete bipartite graph $G$ with parts $X$ and $Y$, the {\em bipartite toughness} of $G$ is defined as $t^{B}(G)=\min\left\{\frac{|S|}{c(G-S)}\right\}$, where the minimum is taken over all proper subsets $S\subset X$ (or $S\subset Y$) such that $c(G-S)>1$. In this paper, we give a sharp spectral radius condition for balanced bipartite graphs $G$ with $t^{B}(G)\geq1$ to guarantee that $G$ contains Hamilton cycles. This solves a problem proposed in \cite{CFL}.