4 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
The smallest nontrivial snarks of oddness 4
The oddness of a cubic graph is the smallest number of odd circuits in a
2-factor of the graph. This invariant is widely considered to be one of the
most important measures of uncolourability of cubic graphs and as such has been
repeatedly reoccurring in numerous investigations of problems and conjectures
surrounding snarks (connected cubic graphs admitting no proper
3-edge-colouring). In [Ars Math. Contemp. 16 (2019), 277-298] we have proved
that the smallest number of vertices of a snark with cyclic connectivity 4 and
oddness 4 is 44. We now show that there are exactly 31 such snarks, all of them
having girth 5. These snarks are built up from subgraphs of the Petersen graph
and a small number of additional vertices. Depending on their structure they
fall into six classes, each class giving rise to an infinite family of snarks
with oddness at least 4 with increasing order. We explain the reasons why these
snarks have oddness 4 and prove that the 31 snarks form the complete set of
snarks with cyclic connectivity 4 and oddness 4 on 44 vertices. The proof is a
combination of a purely theoretical approach with extensive computations
performed by a computer.Comment: 38 pages; submitted for publicatio