31 research outputs found
Cubic graphs with large circumference deficit
The circumference of a graph is the length of a longest cycle. By
exploiting our recent results on resistance of snarks, we construct infinite
classes of cyclically -, - and -edge-connected cubic graphs with
circumference ratio bounded from above by , and
, respectively. In contrast, the dominating cycle conjecture implies
that the circumference ratio of a cyclically -edge-connected cubic graph is
at least .
In addition, we construct snarks with large girth and large circumference
deficit, solving Problem 1 proposed in [J. H\"agglund and K. Markstr\"om, On
stable cycles and cycle double covers of graphs with large circumference, Disc.
Math. 312 (2012), 2540--2544]
Smallest snarks with oddness 4 and cyclic connectivity 4 have order 44
The family of snarks -- connected bridgeless cubic graphs that cannot be
3-edge-coloured -- is well-known as a potential source of counterexamples to
several important and long-standing conjectures in graph theory. These include
the cycle double cover conjecture, Tutte's 5-flow conjecture, Fulkerson's
conjecture, and several others. One way of approaching these conjectures is
through the study of structural properties of snarks and construction of small
examples with given properties. In this paper we deal with the problem of
determining the smallest order of a nontrivial snark (that is, one which is
cyclically 4-edge-connected and has girth at least 5) of oddness at least 4.
Using a combination of structural analysis with extensive computations we prove
that the smallest order of a snark with oddness at least 4 and cyclic
connectivity 4 is 44. Formerly it was known that such a snark must have at
least 38 vertices [J. Combin. Theory Ser. B 103 (2013), 468--488] and one such
snark on 44 vertices was constructed by Lukot'ka et al. [Electron. J. Combin.
22 (2015), #P1.51]. The proof requires determining all cyclically
4-edge-connected snarks on 36 vertices, which extends the previously compiled
list of all such snarks up to 34 vertices [J. Combin. Theory Ser. B, loc.
cit.]. As a by-product, we use this new list to test the validity of several
conjectures where snarks can be smallest counterexamples.Comment: 21 page
Frank number and nowhere-zero flows on graphs
An edge of a graph is called deletable for some orientation if
the restriction of to is a strong orientation. Inspired by a problem
of Frank, in 2021 H\"orsch and Szigeti proposed a new parameter for
-edge-connected graphs, called the Frank number, which refines
-edge-connectivity. The Frank number is defined as the minimum number of
orientations of for which every edge of is deletable in at least one of
them. They showed that every -edge-connected graph has Frank number at most
and that in case these graphs are also -edge-colourable the parameter is
at most . Here we strengthen both results by showing that every
-edge-connected graph has Frank number at most and that every graph
which is -edge-connected and -edge-colourable has Frank number . The
latter also confirms a conjecture by Bar\'at and Bl\'azsik. Furthermore, we
prove two sufficient conditions for cubic graphs to have Frank number and
use them in an algorithm to computationally show that the Petersen graph is the
only cyclically -edge-connected cubic graph up to vertices having Frank
number greater than .Comment: 22 page