It is conjectured by Berge and Fulkerson that every bridgeless cubic graph has six perfect matchings such that each edge is contained in exactly two of them. Hägglund constructed two graphs Blowup(K4, C) and Blowup(Prism,C4). Based on these two graphs, Chen constructed infinite families of bridgeless cubic graphs M0,1,2,...,k−2,k−1 which is obtained from cyclically 4-edge-connected and having a...

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 F of snarks (cyclically 4-edge-connected cubic graph...

If G is a bridgeless cubic graph, Fulkerson conjectured that we can find 6 perfect matchings (a 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 FR-triple is a ...

Algorithmic Aspects of Network Flow: In the previous lecture, we presented the Ford-Fulkerson algorithm. We showed that on termination this algorithm produces the maximum flow in an s-t network. In this lecture we discuss the algorithm’s running time, and discuss more efficient alternatives. Analysis of Ford-Fulkerson: Before discussing the worst-case running time of the Ford-Fulkerson algorith...

If G is a bridgeless cubic graph, Fulkerson conjectured that we can find 6 perfect matchings M1, . . . ,M6 of G with the property that every edge of G is contained in exactly two of them and Berge conjectured that its edge set can be covered by 5 perfect matchings. We define τ(G) as the least number of perfect matchings allowing to cover the edge set of a bridgeless cubic graph and we study thi...

Let G be a cubic graph and the graph 2G is obtained by replacing each edge of G with a pair of parallel edges. A proper 6-edgecoloring of 2G is called a Fulkerson coloring of G. It was conjectured by Fulkerson that every bridgeless cubic graph has a Fulkerson coloring. In this paper we show that for a Petersen-minor free Graph G, G is uniquely Fulkerson colorable if and only if G constructed fr...

The Berge-Fulkerson Conjecture states that every cubic bridgeless graph has six perfect matchings such that every edge of the graph is in exactly two of the perfect matchings. If the Berge-Fulkerson Conjecture is true, then what can we say about the proportion of edges of a cubic bridgeless graph that can be covered by k of its perfect matchings? This is the question we address in this paper. W...

This work presents an algorithm for computing the maximum flow and minimum cut of undirected graphs, based on the well-known algorithm presented by Ford and Fulkerson for directed graphs. The new algorithm is equivalent to just applying Ford and Fulkerson algorithm to the directed graph obtained from original graph but with two directed arcs for each edge in the graph, one in each way. We prese...

It is well known that every cycle of a graph must intersect every cut in an even number of edges. For planar graphs, Ford and Fulkerson proved that, for any edge e, there exists a cycle containing e that intersects every minimal cut containing e in exactly two edges. The main result of this paper generalizes this result to any nonplanar graph G provided G does not have a K 3,3 minor containing ...