Berge-Fulkerson Conjecture Research Articles

Rotation r-graphs

We study rotation r-graphs and show that for every r-graph G of odd regularity there is a simple rotation r-graph G′ such that G can be obtained form G′ by a finite number of 2-cut reductions. As a consequence, some hard conjectures as the (generalized) Berge-Fulkerson Conjecture and Tutte's 3- and 5-flow conjecture can be reduced to rotation r-graphs.

Disjoint odd circuits in a bridgeless cubic graph can be quelled by a single perfect matching

Let G be a bridgeless cubic graph. The Berge–Fulkerson Conjecture (1970s) states that G admits a list of six perfect matchings such that each edge of G belongs to exactly two of these perfect matchings. If answered in the affirmative, two other recent conjectures would also be true: the Fan–Raspaud Conjecture (1994), which states that G admits three perfect matchings such that every edge of G belongs to at most two of them; and a conjecture by Mazzuoccolo (2013), which states that G admits two perfect matchings whose deletion yields a bipartite subgraph of G. It can be shown that given an arbitrary perfect matching of G, it is not always possible to extend it to a list of three or six perfect matchings satisfying the statements of the Fan–Raspaud and the Berge–Fulkerson conjectures, respectively. In this paper, we show that given any 1+-factor F (a spanning subgraph of G such that its vertices have degree at least 1) and an arbitrary edge e of G, there always exists a perfect matching M of G containing e such that G∖(F∪M) is bipartite. Our result implies Mazzuoccolo's conjecture, but not only. It also implies that given any collection of disjoint odd circuits in G, there exists a perfect matching of G containing at least one edge of each circuit in this collection.

Rotation snark, Berge-Fulkerson conjecture and Catlin’s 4-flow reduction

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. An infinite family R, of cyclically 5-edge-connected rotation snarks, was discovered in [European J. Combin. 2021] by Máčajová and Škoviera. In this paper, the Berge-Fulkerson conjecture is verified for the family R, and furthermore, a sup-family of R. Catlin’s contractible configuration and Tutte’s integer flow are applied here as the key methods.

Berge–Fulkerson coloring for some families of superposition snarks

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. This conjecture has been verified for many families of snarks with small (≤5) cyclic edge-connectivity. An infinite family, denoted by SK, of cyclically 6-edge-connected superposition snarks was constructed in [European J. Combin. 2002] by Kochol. In this paper, the Berge–Fulkerson conjecture is verified for the family SK, and, furthermore, some larger families containing SK. This is the first paper about the Berge–Fulkerson conjecture for superposition snarks and cyclically 6-edge-connected snarks. Tutte’s integer flow and Catlin’s contractible configuration are applied here as the key methods.

Connectivity of orientations of 3-edge-connected graphs

We attempt to generalize a theorem of Nash-Williams stating that a graph has a k-arc-connected orientation if and only if it is 2k-edge-connected. In a strongly connected digraph we call an arc deletable if its deletion leaves a strongly connected digraph. Given a 3-edge-connected graph G, we define its Frank number f(G) to be the minimum number k such that there exist k orientations of G with the property that every edge becomes a deletable arc in at least one of these orientations. We are interested in finding a good upper bound for the Frank number. We prove that f(G)≤7 for every 3-edge-connected graph. On the other hand, we show that a Frank number of 3 is attained by the Petersen graph. Further, we prove better upper bounds for more restricted classes of graphs and establish a connection to the Berge–Fulkerson conjecture. We also show that deciding whether all edges of a given subset can become deletable in one orientation is NP-complete.

Reduction of the Berge-Fulkerson conjecture to cyclically 5-edge-connected snarks

The Berge-Fulkerson conjecture, originally formulated in the language of mathematical programming, asserts that the edges of every bridgeless cubic ( 3 3 -valent) graph can be covered with six perfect matchings in such a way that every edge is in exactly two of them. As with several other classical conjectures in graph theory, every counterexample to the Berge-Fulkerson conjecture must be a non- 3 3 -edge-colorable cubic graph. In contrast to Tutte’s 5-flow conjecture and the cycle double conjecture, no nontrivial reduction is known for the Berge-Fulkerson conjecture. In the present paper, we prove that a possible minimum counterexample to the conjecture must be cyclically 5 5 -edge-connected.

Normal 6-edge-colorings of some bridgeless cubic graphs

In an edge-coloring (proper) of a cubic graph, an edge is poor or rich, if the set of colors assigned to the edge and the four edges adjacent it, has exactly three or exactly five distinct colors, respectively. An edge is normal in an edge-coloring if it is rich or poor in this coloring. A normal k-edge-coloring of a cubic graph is an edge-coloring with k colors such that each edge of the graph is normal. We denote by χN′(G) the smallest k, for which G admits a normal k-edge-coloring. Normal edge-colorings were introduced by Jaeger in order to study his well-known Petersen Coloring Conjecture. It is known that proving χN′(G)≤5 for every bridgeless cubic graph is equivalent to proving Petersen Coloring Conjecture. Moreover, Jaeger was able to show that it implies classical conjectures like Cycle Double Cover Conjecture and Berge–Fulkerson Conjecture. Recently, two of the authors were able to show that any simple cubic graph admits a normal 7-edge-coloring, and this result is best possible. In the present paper, we show that any claw-free bridgeless cubic graph, permutation snark, tree-like snark admits a normal 6-edge-coloring. Finally, we show that any bridgeless cubic graph G admits a 6-edge-coloring such that at least 79⋅|E| edges of G are normal.

On two consequences of Berge–Fulkerson conjecture

The classical Berge–Fulkerson conjecture states that any bridgeless cubic graph G admits a list of six perfect matchings such that each edge of G belongs to two of the perfect matchings from the list. In this short note, we discuss two statements that are consequences of this conjecture. The first of them states that for any bridgeless cubic graph G, edge e and i with 0≤i≤2, there are three perfect matchings F1,F2,F3 of G such that F1∩F2∩F3=0̸ and e belongs to exactly i of these perfect matchings. The second one states that for any bridgeless cubic graph G and its vertex v, there are three perfect matchings F1,F2,F3 of G such that F1∩F2∩F3=0̸ and the edges incident to v belong to k1, k2 and k3 of these perfect matchings, where the numbers k1, k2 and k3 satisfy the obvious necessary conditions. In the paper, we show that the first statement is equivalent to Fan–Raspaud conjecture. We also show that the smallest counter-example to the second one is a cyclically 4-edge-connected cubic graph.

Berge–Fulkerson coloring for C(8)‐linked graphs

AbstractIt 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. Let G be a cubic graph and be a 2‐factor of G such that is odd if and only if for some integer k. The 2‐factor F is C(8)‐linked if, for every , there is a circuit of length 8 with edge sequence where and . And the cubic graph G is C(8)‐linked if it contains a C(8)‐linked 2‐factor. It is proved in this article that every C(8)‐linked cubic graph is Berge–Fulkerson colorable. It is also noticed that many classical families of snarks (including some high oddness snarks) are C(8)‐linked. Consequently, the Berge–Fulkerson conjecture is verified for these infinite families of snarks.

Berge–Fulkerson Conjecture on Certain Snarks

A snark is a connected, bridgeless cubic graph with chromatic index equal to 4. The Berge–Fulkerson conjecture proposed in 1971 states that every bridgeless cubic graph contains a family of six perfect matchings such that each edge is contained in exactly two of them. This conjecture holds trivialy for 3-edge colorable graphs. Thus a possible minimum counterexample for the conjecture is a snark. In this paper we shown that the conjecture holds for families of snarks such as Loupekhine snarks of first and second kind and the Watkins snark. We also determine the excessive index for Loupekhine snarks of first and second kind.

On the maximum fraction of edges covered by [formula omitted] perfect matchings in a cubic bridgeless graph

A conjecture of Berge and Fulkerson (1971) states that every cubic bridgeless graph contains 6 perfect matchings covering each edge precisely twice, which easily implies that every cubic bridgeless graph has three perfect matchings with empty intersection (this weaker statement was conjectured by Fan and Raspaud in 1994). Let mt be the supremum of all reals α≤1 such that for every cubic bridgeless graph G, there exist t perfect matchings of G covering a fraction of at least α of the edges of G. It is known that the Berge–Fulkerson conjecture is equivalent to the statement that m5=1, and implies that m4=1415 and m3=45. In the first part of this paper, we show that m4=1415 implies m3=45, and m3=45 implies the Fan–Raspaud conjecture, strengthening a recent result of Tang, Zhang, and Zhu. In the second part of the paper, we prove that for any 2≤t≤4 and for any real τ lying in some appropriate interval, deciding whether a fraction of more than (resp. at least) τ of the edges of a given cubic bridgeless graph can be covered by t perfect matching is an NP-complete problem. This resolves a conjecture of Tang, Zhang, and Zhu.

Perfect matching covering, the Berge–Fulkerson conjecture, and the Fan–Raspaud conjecture

Let mt∗ be the largest rational number such that every bridgeless cubic graph G associated with a positive weight ω has t perfect matchings {M1,…,Mt} with ω(∪i=1tMi)≥mt∗ω(G). It is conjectured in this paper that m3∗=45, m4∗=1415, and m5∗=1, which are called the weighted PM-covering conjectures. The counterparts of this new invariant mt∗ and conjectures for unweighted cubic graphs were introduced by Kaiser et al. (2006). It is observed in this paper that the Berge–Fulkerson conjecture implies the weighted PM-covering conjectures. Each of the weighted PM-covering conjectures is further proved to imply the Fan–Raspaud conjecture. Furthermore, a 3PM-coverage index τ (respectively, τ∗ for the weighted case) is introduced for measuring the maximum ratio of the number of (respectively, the total weight of) edges covered by three perfect matchings in bridgeless cubic graphs and assessing how far a snark is from being 3-edge-colorable. It is proved that the determination of τ∗ for bridgeless cubic graphs is an NP-complete problem.

Flow-continuous mappings—The influence of the group

Many conjectures at the core of graph theory can be formulated as questions about certain group-valued flows: examples include the cycle double-cover conjecture, the Berge–Fulkerson conjecture, and Tutte’s 3-flow, 4-flow, and 5-flow conjectures. As an approach to these problems, Jaeger, and DeVos, Nešetřil, and Raspaud define a notion of graph morphisms continuous with respect to group-valued flows. We discuss the influence of the group on these maps. In particular, we prove that the number of flow-continuous mappings between two graphs does not depend on the group, but only on the largest order of an element of the group (i.e., on the exponent of the group). Further, there is a nice algebraic structure describing for which groups a mapping is flow-continuous.On the combinatorial side, we show that for cubic graphs the only relevant groups are Z2, Z3, and Z.

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

AbstractWe construct a family of r‐graphs having a minimum 1‐factor cover of cardinality (disproving a conjecture of Bonisoli and Cariolaro, Birkhäuser, 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.

New approach to Petersen coloring

Abstract Petersen coloring (defined by Jaeger [On graphic-minimal spaces, Ann. Discrete Math. 8 (1980)]) is a mapping from the edges of a cubic graph to the edges of the Petersen graph, so that three edges adjacent at a vertex are mapped to three edges adjacent at a vertex. The existence of such mapping for every cubic bridgeless graph is known to imply the truth of the Cycle double cover conjecture and of the Berge-Fulkerson conjecture. We develop Jaegerʼs alternate formulation of Petersen coloring in terms of special five-edge colorings. We suggest a weaker conjecture, and provide new techniques to solve it. On a related note, we provide a counterexample to a stronger conjecture by DeVos, Nesetřil, and Raspaud [On edge-maps whose inverse preserves flows and tensions, Graph Theory in Paris, 2006] that asked for an oriented version of Petersen coloring.

A note on Berge–Fulkerson coloring

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 M 1 and M 2 such that (i) M 1 ∪ M 2 induces a 2- regular subgraph of G and (ii) the suppressed graph G ∖ M i ¯ , the graph obtained from G ∖ M i by suppressing all degree-2- vertices, is 3- edge-colorable for each i = 1 , 2 . This lemma is further applied in the verification of Berge–Fulkerson Conjecture for some families of non-3-edge-colorable cubic graphs (such as, Goldberg snarks, flower snarks).

Families of Dot-Product Snarks on Orientable Surfaces of Low Genus

We introduce a generalized dot product and provide some embedding conditions under which the genus of a graph does not rise under a dot product with the Petersen graph. Using these conditions, we disprove a conjecture of Tinsley and Watkins on the genus of dot products of the Petersen graph and show that both Grunbaum’s Conjecture and the Berge-Fulkerson Conjecture hold for certain infinite families of snarks. Additionally, we determine the orientable genus of four known snarks and two known snark families, construct a new example of an infinite family of snarks on the torus, and construct ten new examples of infinite families of snarks on the 2-holed torus; these last constructions allow us to show that there are genus-2 snarks of every even order n ≥ 18.

Abelian Colourings of Cubic Graphs

Abstract We prove that every bridgeless cubic graph G can have its edges properly coloured by non-zero elements of any given Abelian group A of order at least 12 in such a way that at each vertex of G the three colours sum to zero in A. The proof relies on the fact that such colourings depend on certain configurations in Steiner triple systems. In contrast, a similar statement for cyclic groups of order smaller than 10 is false, leaving the problem open only for Z 4 × Z 2 , Z 3 × Z 3 , Z 10 and Z 11 . All the extant cases are closely related to certain conjectures concerning cubic graphs, most notably to the celebrated Berge-Fulkerson Conjecture.