Jaeger's graphs and compatible linear partitions

Jean-Luc Fouquet (1) , Henri Thuillier (1) , Jean-Marie Vanherpe (1) , Adam Pawel Wojda (2)

1 LIFO - Laboratoire d'Informatique Fondamentale d'Orléans
2 AGH UST - AGH University of Science and Technology [Krakow, PL]

A strong matching C in a graph G is a matching C such that there is no edge of E(G) connecting any two edges of C. A cubic graph G is a Jaeger's graph if it contains a perfect matching which is a union of two disjoint strong matchings. We survey here some known results about this family and we give some new results. We defi ne the operation of (L;U)-extension and we show that the family of Jaeger's graphs is generated from some small Jaeger's graphs by using this operation. A linear forest is a graph whose connected components are chordless paths. A linear partition of a graph G is a partition of its edge set into linear forests and la(G) is the minimum number of linear forests in a linear partition. It is well known that la(G) = 2 for any cubic graph G. For a linear partition L = (LB;LR) of G and for each vertex v we de fine e_L(v) as the edge incident to v which is an end edge of a maximal path in LB or LR. We shall say that two linear partitions L = (LB;LR) and L' = (L'B;L'R) are compatible whenever e_L(v) is distinct from e_L' (v) for each vertex v. We show that every Jaeger's graph has two compatible linear partitions and we give some sthrenthening to the conjecture: Every cubic graph has two compatible linear partitions.

Mots clés: linear forest; strong matching; cubic graphs; linear partition

Domaines: Mathématique discrète [cs.DM]

Jean-Luc Fouquet, Henri Thuillier, Jean-Marie Vanherpe, Adam Pawel Wojda. Jaeger's graphs and compatible linear partitions. 2011. ⟨hal-00601554⟩