15 research outputs found
On the excessive [m]-index of a tree
The excessive [m]-index of a graph G is the minimum number of matchings of
size m needed to cover the edge-set of G. We call a graph G [m]-coverable if
its excessive [m]-index is finite. Obviously the excessive [1]-index is |E(G)|
for all graphs and it is an easy task the computation of the excessive
[2]-index for a [2]-coverable graph. The case m=3 is completely solved by
Cariolaro and Fu in 2009. In this paper we prove a general formula to compute
the excessive [4]-index of a tree and we conjecture a possible generalization
for any value of m. Furthermore, we prove that such a formula does not work for
the excessive [4]-index of an arbitrary graph.Comment: 12 pages, 7 figures, to appear in Discrete Applied Mathematic
Excessive [l,m]-factorizations
Given two positive integers l and m, with l 64m, an [l,m]-covering of a graph G is a set M of matchings of G whose union is the edge set of G and such that l 64;|M| 64m for every M. An [l,m]-covering M of G is an excessive [l,m]-factorization of G if the cardinality of M is as small as possible. The number of matchings in an excessive [l,m]-factorization of G (or 1e, if G does not admit an excessive [l,m]-factorization) is a graph parameter called the excessive [l,m]-index of G and denoted by \u3c7[l,m]\u2032(G). In this paper we study such parameter. Our main result is a general formula for the excessive [l,m]-index of a graph G in terms of other graph parameters. Furthermore, we give a polynomial time algorithm which computes \u3c7[l,m]\u2032(G) for any fixed constants l and m and outputs an excessive [l,m]-factorization of G, whenever the latter exists
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
Some snarks are worse than others
Many conjectures and open problems in graph theory can either be reduced to
cubic graphs or are directly stated for cubic graphs. Furthermore, it is known
that for a lot of problems, a counterexample must be a snark, i.e. a bridgeless
cubic graph which is not 3--edge-colourable. In this paper we deal with the
fact that the family of potential counterexamples to many interesting
conjectures can be narrowed even further to the family of
bridgeless cubic graphs whose edge set cannot be covered with four perfect
matchings. The Cycle Double Cover Conjecture, the Shortest Cycle Cover
Conjecture and the Fan-Raspaud Conjecture are examples of statements for which
is crucial. In this paper, we study parameters which have
the potential to further refine and thus enlarge the set of
cubic graphs for which the mentioned conjectures can be verified. We show that
can be naturally decomposed into subsets with increasing
complexity, thereby producing a natural scale for proving these conjectures.
More precisely, we consider the following parameters and questions: given a
bridgeless cubic graph, (i) how many perfect matchings need to be added, (ii)
how many copies of the same perfect matching need to be added, and (iii) how
many 2--factors need to be added so that the resulting regular graph is Class
I? We present new results for these parameters and we also establish some
strong relations between these problems and some long-standing conjectures.Comment: 27 pages, 16 figure