‎The excessive index of a bridgeless cubic graph $G$ is the least integer $k$‎, ‎such that $G$ can be covered by $k$ perfect matchings‎. ‎An equivalent form of Fulkerson conjecture (due to Berge) is that every bridgeless‎ ‎cubic graph has excessive index at most five‎. ‎Clearly‎, ‎Petersen graph is a cyclically 4-edge-connected snark with excessive index at least 5‎, ‎so Fouquet and Vanherpe as...

In this note we construct two infinite snark families which have high oddness and low circumference compared to the number of vertices. Using this construction, we also give a counterexample to a suggested strengthening of Fulkerson’s conjecture by showing that the Petersen graph is not the only cyclically 4-edge connected cubic graph which require at least five perfect matchings to cover its e...

For many of the unsolved problems concerning cycles and matchings in graphs it is known that it is su cient to prove them for snarks, the class of nontrivial 3-regular graphs which cannot be 3-edge coloured. In the rst part of this paper we present a new algorithm for generating all non-isomorphic snarks of a given order. Our implementation of the new algorithm is 14 times faster than previous ...

The Berge–Fulkerson Conjecture states that every cubic bridgeless graph has six perfect matchings such that every edge of the graph is contained in exactly two of these perfect matchings. In this paper, a useful technical lemma is proved that a cubic graph G admits a Berge–Fulkerson coloring if and only if the graph G contains a pair of edge-disjoint matchings M1 and M2 such that (i) M1 ∪ M2 in...

It is widely known that the Ford-Fulkerson procedure for nding the maximumow in a network need not terminate if some of the capacities of the network are irrational. Ford and Fulkerson gave as an example a network with 10 vertices and 48 edges on which their procedure may fail to halt. We construct much smaller and simpler networks on which the same may happen. Our smallest network has only 6 v...

The Fulkerson osteotomy has proved to be a reliable treatment for subluxation of the patella due to malalignment. Aggressive rehabilitation in the early postoperative period is unwise since the proximal tibia is weakened by the oblique osteotomy. Early weight-bearing and unrestricted activity have caused fractures in a few patients. Even late in the postoperative period the osteotomy may advers...