On a family of cubic graphs containing the flower snarks

We consider cubic graphs formed with k ≥ 2 disjoint claws Ci ∼ K1,3 (0 ≤ i ≤ k−1) such that for every integer i modulo k the three vertices of degree 1 of Ci are joined to the three vertices of degree 1 of Ci−1 and joined to the three vertices of degree 1 of Ci+1. Denote by ti the vertex of degree 3 of Ci and by T the set {t1, t2, ..., tk−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 ∪ i=0 V (Ci) \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.