On the Hamiltonian Bicirculants

TL;DR

The paper investigates Hamiltonicity in bicirculant graphs, focusing on the class $B(m;R,S,T)$ and its subfamilies. It develops two concrete constructions, $B(m;a,S,b)$ and $B(m;R,S,T)$, that stitch Hamilton cycles from cyclic Haar subgraphs and from component-wise Hamilton paths, enabling broad Hamiltonian results. The authors prove that all connected bicirculants with at most two spokes ($s\le 2$) are Hamiltonian except $K_2$ and $G(m,2)$ with $m\equiv 5\pmod 6$, and they show that if $m$ is a product of at most three prime powers and $s\ge 3$, then the graph is Hamiltonian; in particular, even $m<210$ and odd $m<1155$ give Hamiltonian bicirculants outside known exceptions. Additional corollaries include that many families, such as those with $d-s$ odd, are Hamiltonian. Collectively, the work advances the Hamiltonicity classification of bicirculants and maps out remaining open cases for further study.

Abstract

A bicirculant is a regular graph that admits a semi-regular automorphism with two vertex-orbits of the same size. By $m$ we denote the size of vertex-orbits and by $d$ the valence of a bicirculant. Furthermore, we denote by $s$ the valence of the bipartite graph joining the two vertex-orbits. In 1983, Brian Alspach proved that the only non-hamiltonian generalized Petersen graphs are $G(m,2)$ with $m \equiv 5 \pmod 6$. In a recent paper we conjectured that this is the only exception among regular, connected bicirculants of degree $d > 1$ and we have verified the conjecture for the quartic bicirculants with $s=2$, also known as the generalized rose window graphs. In this paper we develop tools and apply them for a partial verification of the conjecture. We show that the conjecture holds for all bicirculants with $s \leq 2$. As a consequence we obtain that every connected bicirculant with $s \ge 3$ is hamiltonian if $m$ is a product of at most three prime powers. In particular, every connected bicirculant with $s \ge 3$ is hamiltonian for even $m<210$ and odd $m < 1155$. Our results imply that many other families of bicirculants are hamiltonian. For example, all bicirculants with $d-s$ odd are hamiltonian.

Figures (4)

• Figure 1: A typical bicirculant $B(m;R,S,T) \in {\mathcal{B}}(m;d,s)$, defined by a voltage graph for $d-s$ even (left); for $d-s$ odd a semi-edge is drawn at each vertex (right).
• Figure 2: The construction of a hamilton cycle in $G$ in Case 1 of the proof of Lemma \ref{['pro_method1_case1']}, when $\lambda$ is even, $\mu=0$, and the connected components of $H(m,S)$ are hamiltonian. (a) The bold lines stand for the paths we consider in each component $K_j$, $0\leq j\leq\lambda$. (b) We join the paths through edges of type $a$ -- represented by the thin lines -- as described in the proof of Lemma \ref{['pro_method1_case1']}; the figure shows the case $\lambda=4$. The same construction can be also used when $\lambda$ is odd and $\mu=0$, by replacing the path $u_t\,P'_{0,\lambda}\,u_{\ell}$ with the path $u_0\,P'_{0,\lambda}\,u_p$.
• Figure 3: The construction of a hamilton cycle in $G$ in Case 2 of the proof of Lemma \ref{['pro_method1_case1']} when $\mu>0$; in the figure we have $\lambda=\mu=2$.
• Figure 4: The construction of a hamilton cycle described in Lemma \ref{['pro_construction_Bd1bis']}; the figure shows the case $\lambda=3$. The paths we consider in each component $K_j$, $0\leq j\leq\lambda$, are represented by curved lines, the straight line segments represent the edges of type $a$ connecting the vertices in $K_j$ to the vertices in $K_{j+1}$ that are assigned the same label.

Theorems & Definitions (32)

• Conjecture 1
• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 2.1
• Theorem 2.2
• Theorem 2.3: AlChDe2010
• Theorem 2.4: BoPiZi2025
• Proposition 2.5