On Fulkerson conjecture

If $G$ is a bridgeless cubic graph, Fulkerson conjectured that we can find 6 perfect matchings (a{\em Fulkerson covering}) with the property that every edge of $G$ is contained in exactly two of them. A consequence of the Fulkerson conjecture would be that every bridgeless cubic graph has 3 perfect matchings with empty intersection (this problem is known as the Fan Raspaud Conjecture). A {\em FR-triple} is a set of 3 such perfect matchings. We show here how to derive a Fulkerson covering from two FR-triples. Moreover, we give a simple proof that the Fulkerson conjecture holds true for some classes of well known snarks.Comment: Accepted for publication in Discussiones Mathematicae Graph Theory; Discussiones Mathematicae Graph Theory (2010) xxx-yy

Covering a cubic graph with perfect matchings

Let G be a bridgeless cubic graph. A well-known conjecture of Berge and Fulkerson can be stated as follows: there exist five perfect matchings of G such that each edge of G is contained in at least one of them. Here, we prove that in each bridgeless cubic graph there exist five perfect matchings covering a portion of the edges at least equal to 215/231 . By a generalization of this result, we decrease the best known upper bound, expressed in terms of the size of the graph, for the number of perfect matchings needed to cover the edge-set of G.Comment: accepted for the publication in Discrete Mathematic

On cubic bridgeless graphs whose edge-set cannot be covered by four perfect matchings

The problem of establishing the number of perfect matchings necessary to cover the edge-set of a cubic bridgeless graph is strictly related to a famous conjecture of Berge and Fulkerson. In this paper we prove that deciding whether this number is at most 4 for a given cubic bridgeless graph is NP-complete. We also construct an infinite family $\cal F$ of snarks (cyclically 4-edge-connected cubic graphs of girth at least five and chromatic index four) whose edge-set cannot be covered by 4 perfect matchings. Only two such graphs were known. It turns out that the family $\cal F$ also has interesting properties with respect to the shortest cycle cover problem. The shortest cycle cover of any cubic bridgeless graph with $m$ edges has length at least $\tfrac43m$, and we show that this inequality is strict for graphs of $\cal F$. We also construct the first known snark with no cycle cover of length less than $\tfrac43m+2$.Comment: 17 pages, 8 figure

Covering cubic graphs with matchings of large size

Let m be a positive integer and let G be a cubic graph of order 2n. We consider the problem of covering the edge-set of G with the minimum number of matchings of size m. This number is called excessive [m]-index of G in literature. The case m=n, that is a covering with perfect matchings, is known to be strictly related to an outstanding conjecture of Berge and Fulkerson. In this paper we study in some details the case m=n-1. We show how this parameter can be large for cubic graphs with low connectivity and we furnish some evidence that each cyclically 4-connected cubic graph of order 2n has excessive [n-1]-index at most 4. Finally, we discuss the relation between excessive [n-1]-index and some other graph parameters as oddness and circumference.Comment: 11 pages, 5 figure

Normal edge-colorings of cubic graphs

A normal $k$-edge-coloring of a cubic graph is an edge-coloring with $k$ colors having the additional property that when looking at the set of colors assigned to any edge $e$ and the four edges adjacent it, we have either exactly five distinct colors or exactly three distinct colors. We denote by $\chi'_{N}(G)$ the smallest $k$, for which $G$ admits a normal $k$-edge-coloring. Normal $k$-edge-colorings were introduced by Jaeger in order to study his well-known Petersen Coloring Conjecture. More precisely, it is known that proving $\chi'_{N}(G)\leq 5$ 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 $\chi'_{N}(G)=7$. On the other hand, the known best general upper bound for $\chi'_{N}(G)$ was $9$. Here, we improve it by proving that $\chi'_{N}(G)\leq7$ for any simple cubic graph $G$, which is best possible. We obtain this result by proving the existence of specific no-where zero $\mathbb{Z}_2^2$-flows in $4$-edge-connected graphs.Comment: 17 pages, 6 figure

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 ${\cal S}_{\geq 5}$ 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 ${\cal S}_{\geq 5}$ is crucial. In this paper, we study parameters which have the potential to further refine ${\cal S}_{\geq 5}$ and thus enlarge the set of cubic graphs for which the mentioned conjectures can be verified. We show that ${\cal S}_{\geq 5}$ 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