251 research outputs found
Approximating Vizing's independence number conjecture
In 1965, Vizing conjectured that the independence ratio of edge-chromatic
critical graphs is at most . We prove that for every this conjecture is equivalent to its restriction on a specific set of
edge-chromatic critical graphs with independence ratio smaller than
.Comment: Revised version: The title is change
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
Measures of edge-uncolorability
The resistance of a graph is the minimum number of edges that have
to be removed from to obtain a graph which is -edge-colorable.
The paper relates the resistance to other parameters that measure how far is a
graph from being -edge-colorable. The first part considers regular
graphs and the relation of the resistance to structural properties in terms of
2-factors. The second part studies general (multi-) graphs . Let be
the minimum number of vertices that have to be removed from to obtain a
class 1 graph. We show that , and that this bound is best possible.Comment: 9 page
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