3 research outputs found
Ban--Linial's Conjecture and treelike snarks
A bridgeless cubic graph is said to have a 2-bisection if there exists a
2-vertex-colouring of (not necessarily proper) such that: (i) the colour
classes have the same cardinality, and (ii) the monochromatic components are
either an isolated vertex or an edge. In 2016, Ban and Linial conjectured that
every bridgeless cubic graph, apart from the well-known Petersen graph, admits
a 2-bisection. In the same paper it was shown that every Class I bridgeless
cubic graph admits such a bisection. The Class II bridgeless cubic graphs which
are critical to many conjectures in graph theory are snarks, in particular,
those with excessive index at least 5, that is, whose edge-set cannot be
covered by four perfect matchings. Moreover, Esperet et al. state that a
possible counterexample to Ban--Linial's Conjecture must have circular flow
number at least 5. The same authors also state that although empirical evidence
shows that several graphs obtained from the Petersen graph admit a 2-bisection,
they can offer nothing in the direction of a general proof. Despite some
sporadic computational results, until now, no general result about snarks
having excessive index and circular flow number both at least 5 has been
proven. In this work we show that treelike snarks, which are an infinite family
of snarks heavily depending on the Petersen graph and with both their circular
flow number and excessive index at least 5, admit a 2-bisection.Comment: 10 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
An equivalent formulation of the Fan-Raspaud Conjecture and related problems
In 1994, it was conjectured by Fan and Raspaud that every simple bridgeless cubic
graph has three perfect matchings whose intersection is empty. In this paper we answer
a question recently proposed by Mkrtchyan and Vardanyan, by giving an equivalent formulation of the Fan-Raspaud Conjecture. We also study a possibly weaker conjecture
originally proposed by the first author, which states that in every simple bridgeless cubic
graph there exist two perfect matchings such that the complement of their union is a bipartite graph. Here, we show that this conjecture can be equivalently stated using a variant of
Petersen-colourings, we prove it for graphs having oddness at most four and we give a natural extension to bridgeless cubic multigraphs and to certain cubic graphs having bridges.peer-reviewe