Petersen Cores and the Oddness of Cubic Graphs

Let G be a bridgeless cubic graph. Consider a list of k 1-factors of G. Let be the set of edges contained in precisely i members of the k 1-factors. Let be the smallest over all lists of k 1-factors of G. We study lists by three 1-factors, and call with a -core of G. If G is not 3-edge-colorable, then . In Steffen (J Graph Theory 78 (2015), 195–206) it is shown that if , then is an upper bound for the girth of G. We show that bounds the oddness of G as well. We prove that . If , then every -core has a very specific structure. We call these cores Petersen cores. We show that for any given oddness there is a cyclically 4-edge-connected cubic graph G with . On the other hand, the difference between and can be arbitrarily big. This is true even if we additionally fix the oddness. Furthermore, for every integer , there exists a bridgeless cubic graph G such that .