On isomorphic linear partitions in cubic graphs

A linear forest is a graph whose connected components are chordless paths. A linear partition   of a graph GG is a partition of its edge set into linear forests and la(G)la(G) is the minimum number of linear forests in a linear partition. It is well known that la(G)=2la(G)=2 when GG is a cubic graph and Wormald [N. Wormald, Problem 13, Ars Combinatoria 23(A) (1987) 332–334] conjectured that if |V(G)|≡0|V(G)|≡0 (mod 4), then it is always possible to find a linear partition in two isomorphic linear forests. Here, we give some new results concerning this conjecture.