On odd and semi-odd linear partitions of cubic graphs

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. In this paper we consider linear partitions of cubic simple graphs for which it is well known that la(G) = 2. A linear partition ${\displaystyle L=(L_B,L_R)}$ is said to be odd whenever each path of ${\displaystyle L_B\cup L_R}$ has odd length and semi-odd whenever each path of ${\displaystyle L_B}$ (or each path of ${\displaystyle L_R}$) has odd length. In [2] Aldred and Wormald showed that a cubic graph G is 3-edge colourable if and only if G has an odd linear partition. We give here more precise results and we study moreover relationships between semi-odd linear partitions and perfect matchings.