Hamilton cycles in bicirculants with non-coprime connection sets

arXiv:2611.05531v1 Announce Type: new Abstract: A bicirculant is a regular graph that admits an automorphism having two vertex-orbits of the same size. A bicirculant can be described as follows. Given an integer $m \ge 1$ and sets $R, S, T \subseteq \mathbb{Z}_m$ such that $R=-R$, $T=-T$, $0 \not\in R \cup T$ and $0 \in S$, the graph $B(m;R,S,T)$ has vertex set $V=\{u_0,\dots,u_{m-1},v_0,\dots,v_{m-1}\}$ and edge set $E=\{u_iu_{i+j}| \ i \in \mathbb{Z}_m, j \in R\} \cup \{v_iv_{i+j}| \ i \in \mathbb{Z}_m, j \in T\} \cup \{u_iv_{i+j}| \ i \in \mathbb{Z}_m, j \in S\}.$ Some of the well known bicirculants include the generalized Petersen graphs and Cayley graphs of dihedral groups. In 1983, Brian Alspach proved that the only non-hamiltonian generalized Petersen graphs are $G(m,2)$ where $m \equiv 5 \pmod 6$. In a recent paper we conjectured that this is the only exception among regular connected bicirculants of degree at least two, and we have verified the conjecture for bicirculants with $|S|\le 3$. Results of Alspach, Chen and Dean imply that the conjecture also holds for all bicirculants with $m$ even and $\gcd(m,S)=1$. In this paper, we show that the conjecture holds for all bicirculants with $|S| \geq 4$ and $\gcd(m,S)>1$ such that $R\cup T$ contains an element that is not coprime to $\gcd(m,S)$ and, if $m$ is odd, is also not divisibile by $\gcd(m, S)$ or $R \cap T$ contains an element that is coprime to $\gcd(m,S)$.

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