On a family of cubic graphs containing the flower snarks
Jean‐Luc Fouquet, Henri Thuillier, Jean-Marie Vanherpe · 2009
We consider cubic graphs formed with k ≥ 2 disjoint claws C i ∼ K 1,3 (0 ≤ i ≤ 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 F S(1, k), F S(2, k) and F S(3, k).The graph F S(j, k) (j ∈ {1, 2, 3}) is the graph where the set of vertices ∪ i=k-1 i=0 V (C i ) \ T induce j cycles (note that the graphs F S(2, 2p + 1), p ≥ 2, are the ower snarks dened by Isaacs [8]).We determine the number of perfect matchings of every F S(j, k).A cubic graph G is said to be 2-factor hamiltonian if every 2factor of G is a hamiltonian cycle.We characterize the graphs F S(j, k) that are 2factor hamiltonian (note that F S(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 F S(j, k) that are Jaeger's graphs.