56 research outputs found
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
Snarks with total chromatic number 5
A k-total-coloring of G is an assignment of k colors to the edges and vertices of G, so that adjacent and incident elements have different colors. The total chromatic number of G, denoted by chi(T)(G), is the least k for which G has a k-total-coloring. It was proved by Rosenfeld that the total chromatic number of a cubic graph is either 4 or 5. Cubic graphs with chi(T) = 4 are said to be Type 1, and cubic graphs with chi(T) = 5 are said to be Type 2.
Snarks are cyclically 4-edge-connected cubic graphs that do not allow a 3-edge-coloring. In 2003, Cavicchioli et al. asked for a Type 2 snark with girth at least 5. As neither Type 2 cubic graphs with girth at least 5 nor Type 2 snarks are known, this is taking two steps at once, and the two requirements of being a snark and having girth at least 5 should better be treated independently.
In this paper we will show that the property of being a snark can be combined with being Type 2. We will give a construction that gives Type 2 snarks for each even vertex number n >= 40.
We will also give the result of a computer search showing that among all Type 2 cubic graphs on up to 32 vertices, all but three contain an induced chordless cycle of length 4. These three exceptions contain triangles. The question of the existence of a Type 2 cubic graph with girth at least 5 remains open
Normal 5-edge-coloring of some snarks superpositioned by Flower snarks
An edge e is normal in a proper edge-coloring of a cubic graph G if the
number of distinct colors on four edges incident to e is 2 or 4: A normal
edge-coloring of G is a proper edge-coloring in which every edge of G is
normal. The Petersen Coloring Conjecture is equivalent to stating that every
bridgeless cubic graph has a normal 5-edge-coloring. Since every
3-edge-coloring of a cubic graph is trivially normal, it is suficient to
consider only snarks to establish the conjecture. In this paper, we consider a
class of superpositioned snarks obtained by choosing a cycle C in a snark G and
superpositioning vertices of C by one of two simple supervertices and edges of
C by superedges Hx;y, where H is any snark and x; y any pair of nonadjacent
vertices of H: For such superpositioned snarks, two suficient conditions are
given for the existence of a normal 5-edge-coloring. The first condition yields
a normal 5-edge-coloring for all hypohamiltonian snarks used as superedges, but
only for some of the possible ways of connecting them. In particular, since the
Flower snarks are hypohamiltonian, this consequently yields a normal
5-edge-coloring for many snarks superpositioned by the Flower snarks. The
second sufficient condition is more demanding, but its application yields a
normal 5-edge-colorings for all superpositions by the Flower snarks. The same
class of snarks is considered in [S. Liu, R.-X. Hao, C.-Q. Zhang,
Berge{Fulkerson coloring for some families of superposition snarks, Eur. J.
Comb. 96 (2021) 103344] for the Berge-Fulkerson conjecture. Since we
established that this class has a Petersen coloring, this immediately yields
the result of the above mentioned paper.Comment: 30 pages, 16 figure
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
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