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Discussiones Mathematicae Graph Theory 30(2) (2010) 289-314
DOI: https://doi.org/10.7151/dmgt.1495

ON A FAMILY OF CUBIC GRAPHS CONTAINING THE FLOWER SNARKS

Jean-Luc Fouquet, Henri Thuillier and Jean-Marie Vanherpe

L.I.F.O., Faculté des Sciences, B.P. 6759
Université d'Orléans, 45067 Orléans Cedex 2, France

Abstract

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 {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 ∪_{i = 0}^{i = k-1}V(C_i) ∖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.

Keywords: cubic graph, perfect matching, strong matching, counting, hamiltonian cycle, 2-factor hamiltonian.

2010 Mathematics Subject Classification: Primary 05C70,
Secondary 05C45.

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Received 31 December 2008
Revised 17 September 2009
Accepted 9 November 2009