37 research outputs found
Sandwiching saturation number of fullerene graphs
The saturation number of a graph is the cardinality of any smallest
maximal matching of , and it is denoted by . Fullerene graphs are
cubic planar graphs with exactly twelve 5-faces; all the other faces are
hexagons. They are used to capture the structure of carbon molecules. Here we
show that the saturation number of fullerenes on vertices is essentially
On the Strong Parity Chromatic Number
International audienceA vertex colouring of a 2-connected plane graph G is a strong parity vertex colouring if for every face f and each colour c, the number of vertices incident with f coloured by c is either zero or odd. Czap et al. [Discrete Math. 311 (2011) 512-520] proved that every 2-connected plane graph has a proper strong parity vertex colouring with at most 118 colours. In this paper we improve this upper bound for some classes of plane graphs
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
Three-cuts are a charm: acyclicity in 3-connected cubic graphs
Let be a bridgeless cubic graph. In 2023, the three authors solved a
conjecture (also known as the -Conjecture) made by Mazzuoccolo in 2013:
there exist two perfect matchings of such that the complement of their
union is a bipartite subgraph of . They actually show that given any
-factor (a spanning subgraph of such that its vertices have degree
at least 1) and an arbitrary edge of , there exists a perfect matching
of containing such that is bipartite. This
is a step closer to comprehend better the Fan--Raspaud Conjecture and
eventually the Berge--Fulkerson Conjecture. The -Conjecture, now a
theorem, is also the weakest assertion in a series of three conjectures made by
Mazzuoccolo in 2013, with the next stronger statement being: there exist two
perfect matchings of such that the complement of their union is an acyclic
subgraph of . Unfortunately, this conjecture is not true: Jin, Steffen, and
Mazzuoccolo later showed that there exists a counterexample admitting 2-cuts.
Here we show that, despite of this, every cyclically 3-edge-connected cubic
graph satisfies this second conjecture.Comment: 21 pages, 12 figures. arXiv admin note: text overlap with
arXiv:2204.1002
On the Strong Parity Chromatic Number
International audienceA vertex colouring of a 2-connected plane graph G is a strong parity vertex colouring if for every face f and each colour c, the number of vertices incident with f coloured by c is either zero or odd. Czap et al. [Discrete Math. 311 (2011) 512-520] proved that every 2-connected plane graph has a proper strong parity vertex colouring with at most 118 colours. In this paper we improve this upper bound for some classes of plane graphs
Fractional colorings of cubic graphs with large girth
International audienceWe show that every (sub)cubic n-vertex graph with sufficiently large girth has fractional chromatic number at most 2.2978, which implies that it contains an independent set of size at least 0.4352n. Our bound on the independence number is valid for random cubic graphs as well, as it improves existing lower bounds on the maximum cut in cubic graphs with large girth
Facial parity edge colouring
International audienceA facial parity edge colouring of a connected bridgeless plane graph is an edge colouring in which no two face-adjacent edges (consecutive edges of a facial walk of some face) receive the same colour, in addition, for each face α and each colour c, either no edge or an odd number of edges incident with α is coloured with c. From Vizing's theorem it follows that every 3-connected plane graph has a such colouring with at most Δ* + 1 colours, where Δ* is the size of the largest face. In this paper we prove that any connected bridgeless plane graph has a facial parity edge colouring with at most 92 colours
A superlinear bound on the number of perfect matchings in cubic bridgeless graphs
Lovasz and Plummer conjectured in the 1970's that cubic bridgeless graphs
have exponentially many perfect matchings. This conjecture has been verified
for bipartite graphs by Voorhoeve in 1979, and for planar graphs by Chudnovsky
and Seymour in 2008, but in general only linear bounds are known. In this
paper, we provide the first superlinear bound in the general case.Comment: 54 pages v2: a short (missing) proof of Lemma 10 was adde