148 research outputs found
Petersen cores and the oddness of cubic graphs
Let be a bridgeless cubic graph. Consider a list of 1-factors of .
Let be the set of edges contained in precisely members of the
1-factors. Let be the smallest over all lists of
1-factors of . If is not 3-edge-colorable, then . In
[E. Steffen, 1-factor and cycle covers of cubic graphs, J. Graph Theory 78(3)
(2015) 195-206] it is shown that if , then is
an upper bound for the girth of . We show that bounds the oddness
of 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 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 such that .Comment: 13 pages, 9 figure
- …