Discussiones Mathematicae Graph Theory 29(2) (2009) 275-292
doi: 10.7151/dmgt.1447

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ON ODD AND SEMI-ODD LINEAR PARTITIONS OF CUBIC GRAPHS

Jean-Luc Fouquet, Henri Thuillier, Jean-Marie Vanherpe

L.I.F.O., Faculté des Sciences, B.P. 6759
Université d'Orléans
45067 Orléans Cedex 2, France

Adam P. Wojda

Wydzia Matematyki Stosowanej
Zakad Matematyki Dyskretnej
AGH, Al. Mickiewicza 30, 30-059 Kraków, Poland

Abstract

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 L = (LB,LR) is said to be odd whenever each path of LB∪LR has odd length and semi-odd whenever each path of LB (or each path of LR) 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.

Keywords: Cubic graph, linear arboricity, strong matching, edge-colouring.

2000 Mathematics Subject Classification: Primary 05C70;
Secondary 05C38.

References

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Received 3 December 2007
Revised 13 June 2008
Accepted 13 June 2008