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

[BIBTex] [PDF] [PS]


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


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 = 0i = 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.

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

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


[1] M. Abreu, A.A. Diwan, B. Jackson, D. Labbate and J. Sheehan, Pseudo 2-factor isomorphic regular bipartite graphs, J. Combin. Theory (B) 98 (2008) 432-442, doi: 10.1016/j.jctb.2007.08.006.
[2] S. Bonvicini and G. Mazzuoccolo, On perfectly one-factorable cubic graphs, Electronic Notes in Discrete Math. 24 (2006) 47-51, doi: 10.1016/j.endm.2006.06.008.
[3] J.-L. Fouquet, H. Thuillier, J.-M. Vanherpe and A.P. Wojda, On linear arboricity of cubic graphs, LIFO Univ. d'Orlans - Research Report 13 (2007) 1-28.
[4] J.-L. Fouquet, H. Thuillier, J.-M. Vanherpe and A.P. Wojda, On isomorphic linear partition in cubic graphs, Discrete Math. 309 (2009) 6425-6433, doi: 10.1016/j.disc.2008.10.017.
[5] D.R. Fulkerson, Blocking and anti-blocking pairs of polyhedra, Math. Programming 1 (1971) 168-194, doi: 10.1007/BF01584085.
[6] M. Funk, B. Jackson, D. Labbate and J. Sheehan, 2-factor hamiltonian graphs, J. Combin. Theory (B) 87 (2003) 138-144, doi: 10.1016/S0095-8956(02)00031-X.
[7] M. Funk and D. Labbate, On minimally one-factorable r-regular bipartite graphs, Discrete Math. 216 (2000) 121-137, doi: 10.1016/S0012-365X(99)00241-1.
[8] R. Isaacs, Infinite families of non-trivial trivalent graphs which are not Tait colorable, Am. Math. Monthly 82 (1975) 221-239, doi: 10.2307/2319844.
[9] F. Jaeger, Etude de quelques invariants et problèmes d'existence en théorie de graphes (Thèse d'État, IMAG, Grenoble, 1976).
[10] A. Kotzig, Balanced colourings and the four colour conjecture, in: Proc. Sympos. Smolenice, 1963, Publ. House Czechoslovak Acad. Sci. (Prague, 1964) 63-82.
[11] A. Kotzig, Construction for Hamiltonian graphs of degree three (in Russian), Cas. pest. mat. 87 (1962) 148-168.
[12] A. Kotzig and J. Labelle, Quelques problmes ouverts concernant les graphes fortement hamiltoniens, Ann. Sci. Math. Qubec 3 (1979) 95-106.
[13] D. Labbate, On 3-cut reductions of minimally 1-factorable cubic bigraphs, Discrete Math. 231 (2001) 303-310, doi: 10.1016/S0012-365X(00)00327-7.
[14] D. Labbate, Characterizing minimally 1-factorable r-regular bipartite graphs, Discrete Math. 248 (2002) 109-123, doi: 10.1016/S0012-365X(01)00189-3.
[15] N. Robertson and P. Seymour, Excluded minor in cubic graphs, (announced), see also www.math.gatech.edu/~thomas/OLDFTP/cubic/graphs.
[16] P. Seymour, On multi-colourings of cubic graphs, and conjectures of Fulkerson and Tutte, Proc. London Math. Soc. 38 (1979) 423-460, doi: 10.1112/plms/s3-38.3.423.

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