On a family of cubic graphs containing the flower snarks

TL;DR

This paper introduces a family of cubic graphs constructed from disjoint claws, analyzes their perfect matchings, and characterizes those that are 2-factor hamiltonian or ev, including the flower snarks.

Contribution

It defines three new classes of cubic graphs based on claw arrangements and characterizes their perfect matchings and special properties, extending understanding of flower snarks.

Findings

Number of perfect matchings for each graph class determined

Characterization of 2-factor hamiltonian graphs within the family

Identification of ev graphs among the constructed classes

Abstract

We consider cubic graphs formed with $k \geq 2$ disjoint claws $C_i \sim K_{1, 3}$ ($0 \leq i \leq k-1$) such that for every integer $i$ modulo $k$ the three vertices of degree 1 of $\ C_i$ are joined to the three vertices of degree 1 of $C_{i-1}$ and joined to the three vertices of degree 1 of $C_{i+1}$. Denote by $t_i$ the vertex of degree 3 of $C_i$ and by $T$ the set $\{t_1, t_2,..., t_{k-1}\}$. In such a way we construct three distinct graphs, namely $FS(1,k)$, $FS(2,k)$ and $FS(3,k)$. The graph $FS(j,k)$ ($j \in \{1, 2, 3\}$) is the graph where the set of vertices $\cup_{i=0}^{i=k-1}V(C_i) \setminus T$ induce $j$ cycles (note that the graphs $FS(2,2p+1)$, $p\geq2$, are the flower snarks defined by Isaacs \cite{Isa75}). We determine the number of perfect matchings of every $FS(j,k)$. A cubic graph $G$ is said to be {\em 2-factor hamiltonian} if every 2-factor of $G$ is a hamiltonian cycle. We characterize the graphs $FS(j,k)$ that are 2-factor hamiltonian (note that FS(1,3) is the "Triplex Graph" of Robertson, Seymour and Thomas \cite{RobSey}). A {\em strong matching} $M$ in a graph $G$ is a matching $M$ such that there is no edge of $E(G)$ connecting any two edges of $M$. A cubic graph having a perfect matching union of two strong matchings is said to be a {\em\Jaev}. We characterize the graphs $FS(j,k)$ that are \Jaesv.