On certain edge-transitive bicirculants of twice odd order

TL;DR

This paper characterizes edge-transitive bicirculant graphs of twice odd order, proving that the complement of the Petersen graph is uniquely such a graph with specific properties.

Contribution

It establishes the uniqueness of the complement of the Petersen graph as the only edge-transitive bicirculant of twice odd order with a cycle-induced subgraph and valence at least 6.

Findings

The complement of the Petersen graph is unique under the given conditions.

Identifies specific structural properties of such bicirculants.

Advances understanding of symmetry in special graph classes.

Abstract

A graph admitting an automorphism with two orbits of the same length is called a bicirculant. Recently, Jajcay et al. initiated the investigation of the edge-transitive bicirculants with the properties that one of the subgraphs induced by the latter orbits is a cycle and the valence is at least $6$ (Electron. J. Combin., 2019). We show that the complement of the Petersen graph is the only such graph whose order is twice an odd number.