Hamilton Cycles in Semisymmetric Graphs

TL;DR

This paper investigates Hamilton cycles in semisymmetric graphs, showing that many such graphs of specific orders contain Hamilton cycles, and raises questions about their existence in broader classes.

Contribution

It proves that connected semisymmetric graphs of order 2pq with distinct primes, and certain cubic semisymmetric graphs under 3000 vertices, always contain Hamilton cycles.

Findings

Connected semisymmetric graphs of order 2pq have Hamilton cycles.

Cubic semisymmetric graphs under 3000 vertices have Hamilton cycles.

Open problem: existence of semisymmetric graphs without Hamilton cycles.

Abstract

In light of Lov\'{a}sz's longstanding question on the existence of Hamilton paths in vertex-transitive graphs, this paper considers a natural variant: what if vertex-transitivity is relaxed, yet a high degree of symmetry--specifically edge-transitivity--is retained? To investigate this, we focus on the class of semisymmetric graphs, which are regular, edge-transitive, but not vertex-transitive. In this paper, it will be shown that every connected semisymmetric graph of order $2pq$, where $p$ and $q$ are two distinct primes contains a Hamilton cycle and that every connected cubic semisymmetric graph of order less than 3000 contains a Hamilton cycle too. Based on these observations, the following question is posed: construct a connected semisymmetric graph which has no Hamilton cycle.