71 research outputs found
Some snarks are worse than others
Many conjectures and open problems in graph theory can either be reduced to
cubic graphs or are directly stated for cubic graphs. Furthermore, it is known
that for a lot of problems, a counterexample must be a snark, i.e. a bridgeless
cubic graph which is not 3--edge-colourable. In this paper we deal with the
fact that the family of potential counterexamples to many interesting
conjectures can be narrowed even further to the family of
bridgeless cubic graphs whose edge set cannot be covered with four perfect
matchings. The Cycle Double Cover Conjecture, the Shortest Cycle Cover
Conjecture and the Fan-Raspaud Conjecture are examples of statements for which
is crucial. In this paper, we study parameters which have
the potential to further refine and thus enlarge the set of
cubic graphs for which the mentioned conjectures can be verified. We show that
can be naturally decomposed into subsets with increasing
complexity, thereby producing a natural scale for proving these conjectures.
More precisely, we consider the following parameters and questions: given a
bridgeless cubic graph, (i) how many perfect matchings need to be added, (ii)
how many copies of the same perfect matching need to be added, and (iii) how
many 2--factors need to be added so that the resulting regular graph is Class
I? We present new results for these parameters and we also establish some
strong relations between these problems and some long-standing conjectures.Comment: 27 pages, 16 figure
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
Snark Designs
The main aim of this paper is to solve the design spectrum problem for
Tietze's graph, the two 18-vertex Blanusa snarks, the six snarks on 20 vertices
(including the flower snark J5), the twenty snarks on 22 vertices (including
the two Loupekine snarks) and Goldberg's snark 3. Together with the Petersen
graph (for which the spectrum has already been computed) this list includes all
non-trivial snarks of up to 22 vertices. We also give partial results for a
selection of larger graphs: the two Celmins-Swart snarks, the 26- and 34-vertex
Blanusa snarks, the flower snark J7, the double star snark, Zamfirescu's graph,
Goldberg's snark 5, the Szekeres snark and the Watkins snark.Comment: 92 pages. References updated, pictures added, typos corrected.
Abridged version omitting all design details accepted by Utilitas Mathematic
Generation and Properties of Snarks
For many of the unsolved problems concerning cycles and matchings in graphs
it is known that it is sufficient to prove them for \emph{snarks}, the class of
nontrivial 3-regular graphs which cannot be 3-edge coloured. In the first part
of this paper we present a new algorithm for generating all non-isomorphic
snarks of a given order. Our implementation of the new algorithm is 14 times
faster than previous programs for generating snarks, and 29 times faster for
generating weak snarks. Using this program we have generated all non-isomorphic
snarks on vertices. Previously lists up to vertices have been
published. In the second part of the paper we analyze the sets of generated
snarks with respect to a number of properties and conjectures. We find that
some of the strongest versions of the cycle double cover conjecture hold for
all snarks of these orders, as does Jaeger's Petersen colouring conjecture,
which in turn implies that Fulkerson's conjecture has no small counterexamples.
In contrast to these positive results we also find counterexamples to eight
previously published conjectures concerning cycle coverings and the general
cycle structure of cubic graphs.Comment: Submitted for publication V2: various corrections V3: Figures updated
and typos corrected. This version differs from the published one in that the
Arxiv-version has data about the automorphisms of snarks; Journal of
Combinatorial Theory. Series B. 201
Covering cubic graphs with matchings of large size
Let m be a positive integer and let G be a cubic graph of order 2n. We
consider the problem of covering the edge-set of G with the minimum number of
matchings of size m. This number is called excessive [m]-index of G in
literature. The case m=n, that is a covering with perfect matchings, is known
to be strictly related to an outstanding conjecture of Berge and Fulkerson. In
this paper we study in some details the case m=n-1. We show how this parameter
can be large for cubic graphs with low connectivity and we furnish some
evidence that each cyclically 4-connected cubic graph of order 2n has excessive
[n-1]-index at most 4. Finally, we discuss the relation between excessive
[n-1]-index and some other graph parameters as oddness and circumference.Comment: 11 pages, 5 figure
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 m2mâșÂč. 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
Feedback Numbers of Goldberg Snark, Twisted Goldberg Snarks and Related Graphs
A subset of vertices of a graph G is called a feedback vertex set of G if its removal results in an acyclic subgraph. The minimum cardinality of a feedback vertex set is called the feedback number. In this paper, we determine the exact values of the feedback numbers of the Goldberg snarks Gn and its related graphs Gn*, Twisted Goldberg Snarks TGn and its related graphs TGn*. Let f(n) denote the feedback numbers of these graphs, we prove that f(n)=2n+1, for nâ„3
Graphs, Friends and Acquaintances
As is well known, a graph is a mathematical object modeling the
existence of a certain relation between pairs of elements of a given set.
Therefore, it is not surprising that many of the first results concerning
graphs made reference to relationships between people or groups of
people. In this article, we comment on four results of this kind, which
are related to various general theories on graphs and their applications:
the Handshake lemma (related to graph colorings and Boolean
algebra), a lemma on known and unknown people at a cocktail party
(to Ramsey theory), a theorem on friends in common (to distanceregularity
and coding theory), and Hallâs Marriage theorem (to the
theory of networks). These four areas of graph theory, often with
problems which are easy to state but difficult to solve, are extensively
developed and currently give rise to much research work. As examples
of representative problems and results of these areas, which are
discussed in this paper, we may cite the following: the Four Colors
Theorem (4CTC), the Ramsey numbers, problems of the existence of
distance-regular graphs and completely regular codes, and finally the
study of topological proprieties of interconnection networks.Preprin
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