On a family of cubic graphs containing the flower snarks

Abstract

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 C1 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, ⋯, 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 Ui=k-1V(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.