Spectral analogues of Moon-Moser's theorem on Hamilton paths in bipartite graphs

TL;DR

This paper develops spectral conditions that guarantee Hamilton paths in bipartite graphs, extending classical degree-based theorems by Moon and Moser through spectral graph theory techniques.

Contribution

It introduces spectral analogues of Moon and Moser's theorem, providing new spectral criteria for Hamilton paths in bipartite graphs.

Findings

Spectral conditions ensure Hamilton paths in balanced bipartite graphs.

Structural results relate spectral properties to Hamilton path existence.

Extensions to nearly balanced bipartite graphs are established.

Abstract

In 1962, Erd\H{o}s proved a theorem on the existence of Hamilton cycles in graphs with given minimum degree and number of edges. Significantly strengthening in case of balanced bipartite graphs, Moon and Moser proved a corresponding theorem in 1963. In this paper we establish several spectral analogues of Moon and Moser's theorem on Hamilton paths in balanced bipartite graphs and nearly balanced bipartite graphs. One main ingredient of our proofs is a structural result of its own interest, involving Hamilton paths in balanced bipartite graphs with given minimum degree and number of edges.