Giuseppe Mazzuoccolo

The equivalence of two conjectures of Berge and Fulkerson

Let G be a bridgeless cubic graph. Fulkerson conjectured that there exist six 1-factors of G such that each edge of G is contained in exactly two of them. Berge conjectured that the edge-set of G can be covered with at most five 1-factors. We prove that the two conjectures are equivalent.

Graphs of arbitrary excessive class

We show that there exists a family of r-regular graphs of arbitrarily large excessive index for each integer r greater than 3. Furthermore, we answer a question in Bonisoli and Cariolaro (2007) [1] showing that all the positive integers can be attained as excessive classes of regular graphs.

An Upper Bound for the Excessive Index of an r-Graph

We construct a family of r-graphs having a minimum 1-factor cover of cardinality (disproving a conjecture of Bonisoli and Cariolaro, Birkhauser, Basel, 2007, 73–84). Furthermore, we show the equivalence between the statement that is the best possible upper bound for the cardinality of a minimum 1-factor cover of an r-graph and the well-known generalized Berge–Fulkerson conjecture.

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 article, we prove that deciding whether this number is at most four for a given cubic bridgeless graph is NP-complete. We also construct an infinite family F of snarks cyclically 4-edge-connected cubic graphs of girth at least 5 and chromatic index 4 whose edge-set cannot be covered by four perfect matchings. Only two such graphs were known. It turns out that the family 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 , and we show that this inequality is strict for graphs of F. We also construct the first known snark with no cycle cover of length less than .

On the Automorphic Chromatic Index of a Graph

In this paper we define the automorphic H-chromatic index of a graph Γ as the minimum integer m for which Γ has a proper edge-coloring with m colors which is preserved by a given automorphism group H of Γ. After the description of some properties, we determine upper bounds for this index when H is a cyclic group of prime order. We also show that these upper bounds are best possible in a number of instances.

Covering a cubic graph with perfect matchings

Abstract 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 .

Nowhere-Zero 5-Flows On Cubic Graphs with Oddness 4

Tuttes 5-flow conjecture from 1954 states that every bridgeless graph has a nowhere-zero 5-flow. It suffices to prove the conjecture for cyclically 6-edge-connected cubic graphs. We prove that every cyclically 6-edge-connected cubic graph with oddness at most 4 has a nowhere-zero 5-flow. This implies that every minimum counterexample to the 5-flow conjecture has oddness at least 6.

The structure of graphs with Circular flow number 5 or more, and the complexity of their recognition problem

For some time the Petersen graph has been the only known Snark with circular flow number

On the equitable total chromatic number of cubic graphs

5

Upper Bounds for the Automorphic Chromatic Index of a Graph

(or more, as long as the assertion of Tuttes