7 research outputs found
Perfect Matchings in Claw-free Cubic Graphs
Lovasz and Plummer conjectured that there exists a fixed positive constant c
such that every cubic n-vertex graph with no cutedge has at least 2^(cn)
perfect matchings. Their conjecture has been verified for bipartite graphs by
Voorhoeve and planar graphs by Chudnovsky and Seymour. We prove that every
claw-free cubic n-vertex graph with no cutedge has more than 2^(n/12) perfect
matchings, thus verifying the conjecture for claw-free graphs.Comment: 6 pages, 2 figure
A note on 2--bisections of claw--free cubic graphs
A \emph{--bisection} of a bridgeless cubic graph is a --colouring
of its vertex set such that the colour classes have the same cardinality and
all connected components in the two subgraphs induced by the colour classes
have order at most . Ban and Linial conjectured that {\em every bridgeless
cubic graph admits a --bisection except for the Petersen graph}.
In this note, we prove Ban--Linial's conjecture for claw--free cubic graphs
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
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
Perfect matchings in claw-free cubic graphs
Lovász and Plummer conjectured that there exists a fixed positive constant c such that every cubic n-vertex graph with no cutedge has at least 2 cn perfect matchings. Their conjecture has been verified for bipartite graphs by Voorhoeve and planar graphs by Chudnovsky and Seymour. We prove that every claw-free cubic n-vertex graph with no cutedge has more than 2 n/12 perfect matchings, thus verifying the conjecture for claw-free graphs.