5-cycles and the Petersen graph

We show that if G is a connected bridgeless cubic graph whose every 2-factor is comprised of cycles of length five then G is the Petersen graph. ”The Petersen graph is an obstruction to many properties in graph theory, and often is, or conjectured to be, the only obstruction”. This phrase is taken from one of the series of papers by Robertson, Sanders, Seymour and Thomas that is devoted to the proof of prominent Tutte conjecturea conjecture which states that if the Petersen graph is not a minor of a bridgeless cubic graph G then G is 3-edge-colorable, and which in its turn is a particular case of a much more general conjecture of Tutte stating that every bridgeless graph G has a nowhere zero 4-flow unless the Petersen graph is not a minor of G. Another result that stresses the exceptional role of the Petersen graph is proved by Alspach et al. in [1]. The following striking conjecture of Jaeger states that everything related to the colorings of bridgeless cubic graphs can be reduced to that of the Petersen graph, more specifically, Conjecture 1 Petersen coloring conjecture of Jaeger [4]: the edges of every bridgeless cubic graph G can be mapped into the edges of the Petersen graph in such a way that any three mutually incident edges of G are mapped to three mutually incident edges of the Petersen graph. ∗The author is supported by a grant of Armenian National Science and Education Fund