Hamilton Cycles in Semisymmetric Graphs

Abstract

In light of Lovász'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.

Theorems & Definitions (18)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Definition 2.1
• Proposition 2.2
• Proposition 2.3
• Definition 2.4
• Proposition 2.5
• proof
• Lemma 3.1