32 research outputs found
On the smallest snarks with oddness 4 and connectivity 2
A snark is a bridgeless cubic graph which is not 3-edge-colourable. The
oddness of a bridgeless cubic graph is the minimum number of odd components in
any 2-factor of the graph.
Lukot'ka, M\'acajov\'a, Maz\'ak and \v{S}koviera showed in [Electron. J.
Combin. 22 (2015)] that the smallest snark with oddness 4 has 28 vertices and
remarked that there are exactly two such graphs of that order. However, this
remark is incorrect as -- using an exhaustive computer search -- we show that
there are in fact three snarks with oddness 4 on 28 vertices. In this note we
present the missing snark and also determine all snarks with oddness 4 up to 34
vertices.Comment: 5 page
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
An exploration of two infinite families of snarks
Thesis (M.S.) University of Alaska Fairbanks, 2019In this paper, we generalize a single example of a snark that admits a drawing with even rotational symmetry into two infinite families using a voltage graph construction techniques derived from cyclic Pseudo-Loupekine snarks. We expose an enforced chirality in coloring the underlying 5-pole that generated the known example, and use this fact to show that the infinite families are in fact snarks. We explore the construction of these families in terms of the blowup construction. We show that a graph in either family with rotational symmetry of order m has automorphism group of order m2m⁺¹. The oddness of graphs in both families is determined exactly, and shown to increase linearly with the order of rotational symmetry.Chapter 1: Introduction -- 1.1 General Graph Theory -- Chapter 2: Introduction to Snarks -- 2.1 History -- 2.2 Motivation -- 2.3 Loupekine Snarks and k-poles -- 2.4 Conditions on Triviality -- Chapter 3: The Construction of Two Families of Snarks -- 3.1 Voltage Graphs and Lifts -- 3.2 The Family of Snarks, Fm -- 3.3 A Second Family of Snarks, Rm -- Chapter 4: Results -- 4.1 Proof that the graphs Fm and Rm are Snarks -- 4.2 Interpreting Fm and Rm as Blowup Graphs -- 4.3 Automorphism Group -- 4.4 Oddness -- Chapter 5: Conclusions and Open Questions -- References
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