On a family of cubic graphs containing the flower snarks

We consider cubic graphs formed with k ≥ 2 disjoint claws ${\displaystyle C_i\sim K_{1,3}}$ (0 ≤ i ≤ k-1) such that for every integer i modulo k the three vertices of degree 1 of ${\displaystyle C_i}$ are joined to the three vertices of degree 1 of ${\displaystyle C_{i-1}}$ and joined to the three vertices of degree 1 of ${\displaystyle C_{i+1}}$. Denote by ${\displaystyle t_i}$ the vertex of degree 3 of ${\displaystyle C_i}$ and by T the set ${\displaystyle \{t₁,t₂,...,t_{k-1}\}}$. 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 ∈ {1,2,3}) is the graph where the set of vertices ${\displaystyle {\bigcup }_{i=0}^{i=k-1}V\left(C_i\right)\setminus T}$ induce j cycles (note that the graphs FS(2,2p+1), p ≥ 2, are the flower snarks defined by Isaacs [8]). We determine the number of perfect matchings of every FS(j,k). A cubic graph G is said to be 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 [15]). A 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 Jaeger's graph. We characterize the graphs FS(j,k) that are Jaeger's graphs.