On isomorphic linear partitions in cubic graphs

A linear forest is a graph that 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 l a ( G ) = 2 when G is a cubic graph and Wormald [Wormald, N., Problem 13, Ars Combinatoria 23 (1987), pp. 332–334] conjectured that if | V ( G ) | ≡ 0 ( mod 4 ) , then it is always possible to find a linear partition in two isomorphic linear forests. We give here some new results concerning this conjecture.