Berge's Conjecture Research Articles

Unions of perfect matchings in r-graphs

An r-regular graph is said to be an r-graph if |∂(X)|≥r for each odd set X⊆V(G), where |∂(X)| denotes the set of edges with precisely one end in X. Note that every connected bridgeless cubic graph is a 3-graph. The Berge Conjecture states that every 3-graph G has five perfect matchings such that each edge of G is contained in at least one of them. Likewise, generalization of the Berge Conjecture asserts that every r-graph G has 2r−1 perfect matchings that covers each e∈E(G) at least once. A natural question to ask in the light of the Generalized Berge Conjecture is that what can we say about the proportion of edges of an r-graph that can be covered by union of t perfect matchings? In this paper we provide a lower bound to this question. We will also present a new conjecture that might help towards the proof of the Generalized Berge Conjecture.

The complete short proof of the Berge conjecture

<p>We say that a graph <em>B</em> is berge if every graph <em>B'</em> ∈ {<em>B</em>,<em>B̄</em><em></em>} does not contain an induced cycle of odd length ≥ 5 [<span><em>B̄</em></span> is the complementary graph of <em>B</em>}.</p><p>A graph G is perfect if every induced subgraph <em>G'</em> of <em>G</em> satisfies <em>χ</em>(<em>G'</em>)=<em>ω</em>(<em>G'</em>), where <em>χ</em>(<em>G'</em>) is the chromatic number of <em>G'</em> and <em>ω</em>(<em>G'</em>) is the clique number of <em>G'</em>. The Berge conjecture states that a graph <em>H</em> is perfect if and only if <em>H</em> is berge. Indeed, the Berge problem (or the difficult part of the Berge conjecture) consists to show that <em>χ</em>(<em>B</em>)=<em>ω</em>(<em>B</em>) for every berge graph <em>B</em>. In this paper, we give the direct short proof of the Berge conjecture by reducing the Berge problem into a simple equation of three unknowns and by using trivial complex calculus coupled with elementary computation and a trivial reformulation of that problem via the reasoning by reduction to absurd [we recall that the Berge conjecture was first proved by Chudnovsky, Robertson, Seymour and Thomas in a paper of at least 143 pages long. That being said, the new proof given in this paper is far more easy and more short].</p><p>Our work in this paper is original and is completely different from all strong investigations made by Chudnovsky, Robertson, Seymour and Thomas in their manuscript of at least 143 pages long.</p>

A Class of Cubic Graphs Satisfying Berge Conjecture

A Class of Cubic Graphs Satisfying Berge Conjecture

Perfect Matching Covers of Cubic Graphs of Oddness 2

A perfect matching cover of a graph $G$ is a set of perfect matchings of $G$ such that each edge of $G$ is contained in at least one member of it. Berge conjectured that every bridgeless cubic graph has a perfect matching cover of order at most 5. The Berge Conjecture is largely open and it is even unknown whether a constant integer $c$ does exist such that every bridgeless cubic graph has a perfect matching cover of order at most $c$. In this paper, we show that a bridgeless cubic graph $G$ has a perfect matching cover of order at most 11 if $G$ has a 2-factor in which the number of odd circuits is 2.

Covering a cubic graph by 5 perfect matchings

Berge Conjecture states that every bridgeless cubic graph has 5 perfect matchings such that each edge is contained in at least one of them. In this paper, we show that Berge Conjecture holds for two classes of cubic graphs, cubic graphs with a circuit missing only one vertex and bridgeless cubic graphs having a 2-factor with exactly two circuits. The first part of this result implies that Berge Conjecture holds for hypohamiltonian cubic graphs.

Knot commensurability and the Berge conjecture

We investigate commensurability classes of hyperbolic knot complements in the generic case of knots without hidden symmetries. We show that such knot complements which are commensurable are cyclically commensurable, and that there are at most $3$ hyperbolic knot complements in a cyclic commensurability class. Moreover if two hyperbolic knots have cyclically commensurable complements, then they are fibered with the same genus and are chiral. A characterisation of cyclic commensurability classes of complements of periodic knots is also given. In the non-periodic case, we reduce the characterisation of cyclic commensurability classes to a generalization of the Berge conjecture.

A unified approach to known and unknown cases of Berge's conjecture

AbstractBerge's elegant dipath partition conjecture from 1982 states that in a dipath partition P of the vertex set of a digraph minimizing , there exists a collection Ck of k disjoint independent sets, where each dipath P∈P meets exactly min{|P|, k} of the independent sets in C. This conjecture extends Linial's conjecture, the Greene–Kleitman Theorem and Dilworth's Theorem for all digraphs. The conjecture is known to be true for acyclic digraphs. For general digraphs, it is known for k=1 by the Gallai–Milgram Theorem, for k⩾λ (where λis the number of vertices in the longest dipath in the graph), by the Gallai–Roy Theorem, and when the optimal path partition P contains only dipaths P with |P|⩾k. Recently, it was proved (Eur J Combin (2007)) for k=2.There was no proof that covers all the known cases of Berge's conjecture. In this article, we give an algorithmic proof of a stronger version of the conjecture for acyclic digraphs, using network flows, which covers all the known cases, except the case k=2, and the new, unknown case, of k=λ−1 for all digraphs. So far, there has been no proof that unified all these cases. This proof gives hope for finding a proof for all k.

Knots which admit a surgery with simple knot Floer homology groups

The importance of studying knots inside rational homology spheres which have simple knot Floer homology came up in the study of the Berge conjecture using techniques from Heegaard Floer homology by Hedden [7], Rasmussen [14] and Baker, Grigsby and Hedden [1]. By definition, a knot K inside a rational homology sphere X has simple knot Floer homology if the rank of b HFK.X;K/ is equal to the rank of bHF.X / (see Oszvath and Szabo [11; 10] and Rasmussen [15] for the background on Heegaard Floer homology and knot Floer homology). The Berge conjecture concerning knots in S which admit a lens space surgery may almost be reduced to showing that a knot inside a lens space with simple knot Floer homology is simple.

An Original Speech Around the Difficult Part of the Berge Conjecture (Solved by Chudnovsky, Robertson, Seymour and Thomas)

The Berge conjecture was proven by Chudnovsky, Robertson, Seymour and Thomas in a paper of 146 pages long (see [1]); in this manuscript, via an original speech and simple results, we rigorously simplify the understanding of this solved conjecture. It will appear that what Chudnovsky, Robertson, Seymour and Thomas were proved in their paper of 146 pages long, was an analytic conjecture stated in a very small class of graphs. We say that a graph $B$ is berge (see [4]) if every graph $B'\in \{ B, \bar{B}\}$ does not contain an induced cycle of odd length $\geq 5$ ($\bar{B}$ is the complementary graph of $B$). A graph $G$ is perfect if every induced subgraph $G'$ of $G$ satisfies $\chi(G')=\omega(G')$, where $\chi(G')$ is the chromatic number of $G'$ and $\omega(G')$ is the clique number of $G'$. The Berge conjecture states that a graph $H$ is perfect if and only if $H$ is berge. Indeed, the difficult part of the Berge conjecture consists to show that $\chi(B)=\omega(B)$ for every berge graph $B$ (Briefly, the difficult part of the Berge conjecture will be called the Berge problem (see [4])). The Hadwiger conjecture (see [4]), states that every graph $G$ satisfies $\chi(G) \leq \eta(G)$ (where $\eta(G)$ is the hadwiger number of $G$ (i.e. the maximum of $p$ such that $G$ is contractible to the complete graph $K_{p}$)). In [4], it is presented an original investigation around the Hadwiger conjecture and the Berge problem. More precisely, in [4], via two simple Theorems, it is shown that the Hadwiger conjecture and the Berge problem are curiously resembling, so resembling that they seem identical (indeed, they can be restated in ways that resemble each other). In this paper, via only original speech and results, we rigorously simplify the understanding of the Berge problem. Moreover, it will appear that what Chudnovsky, Robertson, Seymour and Thomas were proved in their manuscript of 146 pages long, was an analytic conjecture stated in a very small class of graphs.

On Floer homology and the Berge conjecture on knots admitting lens space surgeries

We complete the first step in a two-part program proposed by Baker, Grigsby, and the author to prove that Berge’s construction of knots in the three-sphere which admit lens space surgeries is complete. The first step, which we prove here, is to show that a knot in a lens space with a threesphere surgery has simple (in the sense of rank) knot Floer homology. The second (conjectured) step involves showing that, for a fixed lens space, the only knots with simple Floer homology belong to a simple finite family. Using results of Baker, we provide evidence for the conjectural part of the program by showing that it holds for a certain family of knots. Coupled with work of Ni, these knots provide the first infinite family of non-trivial knots which are characterized by their knot Floer homology. As another application, we provide a Floer homology proof of a theorem of Berge.

A simple dissertation around the Berge problem and the Hadwiger conjecture

The Hadwiger conjecture (recall (see [8]) that the famous four-color problem is a special case of the Hadwiger conjecture) states that every graph G satisfies χ(G) ≤ η(G) (where χ(G) is the chromatic number of G, and η(G) is the hadwiger number of G (i.e. the maximum of p such that G is contractible to the complete graph Kp )). The Berge problem was proved by Chudnovsky, Robertson, Seymour and Thomas in a paper of 146 pages long (see [1]). In this manuscript, via an original speech and results, we rigorously simplify the understanding of the Berge problem and the Hadwiger conjecture. It will appear that what Chudnovsky, Robertson, Seymour and Thomas were proved in their paper of 146 pages long, was an analytic conjecture stated in a very small class of graphs; it will also appear that to solve the famous Hadwiger conjecture is equivalent to solve an analytic conjecture stated on a very small class of graphs. Moreover, we will see that the Hadwiger conjecture and the Berge problem are rigorously resembling. Based on this resemblance, it will become natural and not surprising to us to conjecture that the Hadwiger conjecture and the Berge conjecture are strongly related.

Grid diagrams and Legendrian lens space links

Grid diagrams encode useful geometric information about knots in S^3. In particular, they can be used to combinatorially define the knot Floer homology of a knot K in S^3, and they have a straightforward connection to Legendrian representatives of K in (S^3, \xi_\st), where \xi_\st is the standard, tight contact structure. The definition of a grid diagram was extended to include a description for links in all lens spaces, resulting in a combinatorial description of the knot Floer homology of a knot K in L(p, q) for all p > 0. In the present article, we explore the connection between lens space grid diagrams and the contact topology of a lens space. Our hope is that an understanding of grid diagrams from this point of view will lead to new approaches to the Berge conjecture, which claims to classify all knots in S^3 upon which surgery yields a lens space.

Grid Diagrams for Lens Spaces and Combinatorial Knot Floer Homology

Similar to knots in S^3, any knot in a lens space has a grid diagram from which one can combinatorially compute all of its knot Floer homology invariants. We give an explicit description of the generators, differentials, and rational Maslov and Alexander gradings in terms of combinatorial data on the grid diagram. Motivated by existing results for the Floer homology of knots in S^3 and the similarity of the combinatorics presented here, we conjecture that a certain family of knots is characterized by their Floer homology. Coupled with work of the third author, an affirmative answer to this would prove the Berge conjecture, which catalogs the knots in S^3 admitting lens space surgeries.

Berge's conjecture on directed path partitions—a survey

Berge's conjecture from 1982 on path partitions in directed graphs generalizes and extends Dilworth's theorem and the Greene–Kleitman theorem which are well known for partially ordered sets. The conjecture relates path partitions to a collection of k independent sets, for each k ⩾ 1 . The conjecture is still open and intriguing for all k > 1 . 1 1 Only recently it was proved Berger and Ben-Arroyo Hartman [56] for k = 2 (added in proof). In this paper, we will survey partial results on the conjecture, look into different proof techniques for these results, and relate the conjecture to other theorems, conjectures and open problems of Berge and other mathematicians.

Recursive generation of partitionable graphs

AbstractResults of Lovász (1972) and Padberg (1974) imply that partitionable graphs contain all the potential counterexamples to Berge's famous Strong Perfect Graph Conjecture. A recursive method of generating partitionable graphs was suggested by Chvátal, Graham, Perold, and Whitesides (1979). Results of Sebő (1996) entail that Berge's conjecture holds for all the partitionable graphs obtained by this method. Here we suggest a more general recursion. Computer experiments show that it generates all the partitionable graphs with ω=3,α ≤ 9 (and we conjecture that the same will hold for bigger α, too) and many but not all for (ω,α)=(4,4) and (4,5). Here, α and ω are respectively the clique and stability numbers of a partitionable graph, that is the numbers of vertices in its maximum cliques and stable sets. All the partitionable graphs generated by our method contain a critical ω‐clique, that is an ω‐clique which intersects only 2ω−2 other ω‐cliques. This property might imply that in our class there are no counterexamples to Berge's conjecture (cf. Sebő (1996)), however this question is still open. © 2002 Wiley Periodicals, Inc. J Graph Theory 41: 259–285, 2002

On the edge-coloring property for Hanani triple systems

We prove that the closure of every Hanani triple system is a resolvable hypergraph, hence it satisfies the Berge's conjecture on linear hypergraphs.

On minimal imperfect graphs with circular symmetry

Results of Lovász and Padberg entail that the class of so-called partitionable graphs contains all the potential counterexamples to Berge's famous Strong Perfect Graph Conjecture, which asserts that the only minimal imperfect graphs are the odd chordless cycles with at least five vertices (“odd holes”) and their complements (“odd antiholes”). Only two constructions (due to Chvátal, Graham, Perold, and Whitesides) are known for making partitionable graphs. The first one does not produce any counterexample to Berge's Conjecture, as shown by Sebö. Here we prove that the second one does not produce any counterexample either. This second construction is given by near-factorizations of cyclic groups based on the so-called “British number systems” introduced by De Bruijn. All the partitionable graphs generated by this second construction (in particular odd holes and odd antiholes) have circular symmetry. No other partitionable graph with circular symmetry is known, and we conjecture that none exists; in this direction we prove that any counterexample must contain a clique and a stable set with at least six vertices each. Also we prove that every minimal imperfect graph with circular symmetry must have an odd number of vertices. © 1998 John Wiley & Sons, Inc. J Graph Theory 29: 209–225, 1998

On minimal imperfect graphs with circular symmetry

Results of Lovász and Padberg entail that the class of so-called partitionable graphs contains all the potential counterexamples to Berge's famous Strong Perfect Graph Conjecture, which asserts that the only minimal imperfect graphs are the odd chordless cycles with at least five vertices (“odd holes”) and their complements (“odd antiholes”). Only two constructions (due to Chvátal, Graham, Perold, and Whitesides) are known for making partitionable graphs. The first one does not produce any counterexample to Berge's Conjecture, as shown by Sebö. Here we prove that the second one does not produce any counterexample either. This second construction is given by near-factorizations of cyclic groups based on the so-called “British number systems” introduced by De Bruijn. All the partitionable graphs generated by this second construction (in particular odd holes and odd antiholes) have circular symmetry. No other partitionable graph with circular symmetry is known, and we conjecture that none exists; in this direction we prove that any counterexample must contain a clique and a stable set with at least six vertices each. Also we prove that every minimal imperfect graph with circular symmetry must have an odd number of vertices. © 1998 John Wiley & Sons, Inc. J Graph Theory 29: 209–225, 1998