32 research outputs found
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
On cubic bridgeless graphs whose edge-set cannot be covered by four perfect matchings
The problem of establishing the number of perfect matchings necessary to
cover the edge-set of a cubic bridgeless graph is strictly related to a famous
conjecture of Berge and Fulkerson. In this paper we prove that deciding whether
this number is at most 4 for a given cubic bridgeless graph is NP-complete. We
also construct an infinite family of snarks (cyclically
4-edge-connected cubic graphs of girth at least five and chromatic index four)
whose edge-set cannot be covered by 4 perfect matchings. Only two such graphs
were known. It turns out that the family also has interesting
properties with respect to the shortest cycle cover problem. The shortest cycle
cover of any cubic bridgeless graph with edges has length at least
, and we show that this inequality is strict for graphs of .
We also construct the first known snark with no cycle cover of length less than
.Comment: 17 pages, 8 figure
Normal edge-colorings of cubic graphs
A normal -edge-coloring of a cubic graph is an edge-coloring with
colors having the additional property that when looking at the set of colors
assigned to any edge and the four edges adjacent it, we have either exactly
five distinct colors or exactly three distinct colors. We denote by
the smallest , for which admits a normal
-edge-coloring. Normal -edge-colorings were introduced by Jaeger in order
to study his well-known Petersen Coloring Conjecture. More precisely, it is
known that proving for every bridgeless cubic graph is
equivalent to proving Petersen Coloring Conjecture and then, among others,
Cycle Double Cover Conjecture and Berge-Fulkerson Conjecture. Considering the
larger class of all simple cubic graphs (not necessarily bridgeless), some
interesting questions naturally arise. For instance, there exist simple cubic
graphs, not bridgeless, with . On the other hand, the known
best general upper bound for was . Here, we improve it by
proving that for any simple cubic graph , which is best
possible. We obtain this result by proving the existence of specific no-where
zero -flows in -edge-connected graphs.Comment: 17 pages, 6 figure
Normal 6-edge-colorings of some bridgeless cubic graphs
In an edge-coloring of a cubic graph, an edge is poor or rich, if the set of
colors assigned to the edge and the four edges adjacent it, has exactly five or
exactly three distinct colors, respectively. An edge is normal in an
edge-coloring if it is rich or poor in this coloring. A normal
-edge-coloring of a cubic graph is an edge-coloring with colors such
that each edge of the graph is normal. We denote by the smallest
, for which admits a normal -edge-coloring. Normal edge-colorings
were introduced by Jaeger in order to study his well-known Petersen Coloring
Conjecture. It is known that proving for every bridgeless
cubic graph is equivalent to proving Petersen Coloring Conjecture. Moreover,
Jaeger was able to show that it implies classical conjectures like Cycle Double
Cover Conjecture and Berge-Fulkerson Conjecture. Recently, two of the authors
were able to show that any simple cubic graph admits a normal
-edge-coloring, and this result is best possible. In the present paper, we
show that any claw-free bridgeless cubic graph, permutation snark, tree-like
snark admits a normal -edge-coloring. Finally, we show that any bridgeless
cubic graph admits a -edge-coloring such that at least edges of are normal.Comment: 17 pages, 11 figures. arXiv admin note: text overlap with
arXiv:1804.0944
The Cost of Perfection for Matchings in Graphs
Perfect matchings and maximum weight matchings are two fundamental
combinatorial structures. We consider the ratio between the maximum weight of a
perfect matching and the maximum weight of a general matching. Motivated by the
computer graphics application in triangle meshes, where we seek to convert a
triangulation into a quadrangulation by merging pairs of adjacent triangles, we
focus mainly on bridgeless cubic graphs. First, we characterize graphs that
attain the extreme ratios. Second, we present a lower bound for all bridgeless
cubic graphs. Third, we present upper bounds for subclasses of bridgeless cubic
graphs, most of which are shown to be tight. Additionally, we present tight
bounds for the class of regular bipartite graphs
Berge - Fulkerson Conjecture And Mean Subtree Order
Let be a graph, and be the vertex set and edge set of , respectively. A perfect matching of is a set of edges, , such that each vertex in is incident with exactly one edge in . An -regular graph is said to be an -graph if for each odd set , where denotes the set of edges with precisely one end in . One of the most famous conjectures in Matching Theory, due to Berge, states that every 3-graph has five perfect matchings such that each edge of is contained in at least one of them. Likewise, generalization of the Berge Conjecture given, by Seymour, asserts that every -graph has perfect matchings that covers each at least once. In the first part of this thesis, I will provide a lower bound to number of perfect matchings needed to cover the edge set of an -graph. I will also present some new conjectures that might shade a light towards the generalized Berge conjecture. In the second part, I will present a proof of a conjecture stating that there exists a pair of graphs and with , and such that mean subtree order of is smaller then mean subtree order of
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
Circuits, Perfect Matchings and Paths in Graphs
We primarily consider the problem of finding a family of circuits to cover a bidgeless graph (mainly on cubic graph) with respect to a given weight function defined on the edge set. The first chapter of this thesis is going to cover all basic concepts and notations will be used and a survey of this topic.;In Chapter two, we shall pay our attention to the Strong Circuit Double Cover Conjecture (SCDC Conjecture). This conjecture was verified for some graphs with special structure. As the complement of two factor in cubic graph, the Berge-Fulkersen Conjecture was introduced right after SCDC Conjecture. In Chapter three, we shall present a series of conjectures related to perfect matching covering and point out their relationship.;In last chapter, we shall introduce the saturation number, in contrast to extremal number (or known as Turan Number), and describe the edge spectrum of saturation number for small paths, where the spectrum was consisted of all possible integers between saturation number and Turan number