M-alternating Hamilton paths and M-alternating Hamilton cycles

TL;DR

This paper investigates conditions under which graphs contain $M$-alternating Hamilton paths and cycles, providing new criteria based on degree sums and connectivity, with implications for bipartite and extendable graphs.

Contribution

The paper introduces new sufficient conditions for the existence of $M$-alternating Hamilton paths and cycles in graphs, extending previous results and linking graph connectivity with alternating cycle existence.

Findings

In bipartite graphs, degree sum conditions guarantee $M$-alternating Hamilton cycles.

High connectivity ensures the existence of $M$-alternating Hamilton cycles or classifies exceptional cases.

Results apply to $k$-extendable graphs, establishing conditions for $M$-alternating Hamilton cycles.

Abstract

We study $M$-alternating Hamilton paths and $M$-alternating Hamilton cycles in a simple connected graph $G$ on $\nu$ vertices with a perfect matching $M$. Let $G$ be a bipartite graph, we prove that if for any two vertices $x$ and $y$ in different parts of $G$, $d(x)+d(y)\geq \nu/2+2$, then $G$ has an $M$-alternating Hamilton cycle. For general graphs, a condition for the existence of an $M$-alternating Hamilton path starting and ending with edges in $M$ is put forward. Then we prove that if $\kappa(G)\geq\nu/2$, where $\kappa(G)$ denotes the connectivity of $G$, then $G$ has an $M$-alternating Hamilton cycle or belongs to one class of exceptional graphs. Lou and Yu \cite{LY} have proved that every $k$-extendable graph $H$ with $k\geq\nu/4$ is bipartite or satisfies $\kappa(H)\geq 2k$. Combining this result with those we obtain we prove the existence of $M$-alternating Hamilton cycles in $H$.