5 research outputs found
Colouring of cubic graphs by Steiner triple systems
Let S be a Steiner triple system and G a cubic graph. We say that G is S-colourable if its edges can be coloured so that at each vertex the incident colours form a triple of S. We show that if S is a projective system PG(n, 2), n >= 2, then G is S-colourable if and only if it is bridgeless, and that every bridgeless cubic graph has an S-colouring for every Steiner triple system of order greater than 3. We establish a condition on a cubic graph with a bridge which ensures that it fails to have an S-colouring if S is an affine system, and we conjecture that this is the only obstruction to colouring any cubic graph with any non-projective system of order greater than 3
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
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
Disjoint odd circuits in a bridgeless cubic graph can be quelled by a single perfect matching
Let be a bridgeless cubic graph. The Berge-Fulkerson Conjecture (1970s)
states that admits a list of six perfect matchings such that each edge of
belongs to exactly two of these perfect matchings. If answered in the
affirmative, two other recent conjectures would also be true: the Fan-Raspaud
Conjecture (1994), which states that admits three perfect matchings such
that every edge of belongs to at most two of them; and a conjecture by
Mazzuoccolo (2013), which states that admits two perfect matchings whose
deletion yields a bipartite subgraph of . It can be shown that given an
arbitrary perfect matching of , it is not always possible to extend it to a
list of three or six perfect matchings satisfying the statements of the
Fan-Raspaud and the Berge-Fulkerson conjectures, respectively. In this paper,
we show that given any -factor (a spanning subgraph of such that
its vertices have degree at least 1) and an arbitrary edge of , there
always exists a perfect matching of containing such that
is bipartite. Our result implies Mazzuoccolo's
conjecture, but not only. It also implies that given any collection of disjoint
odd circuits in , there exists a perfect matching of containing at least
one edge of each circuit in this collection.Comment: 13 pages, 8 figure