The bunkbed conjecture (also spelled bunk bed conjecture) is a statement in percolation theory, a branch of mathematics that studies the behavior of connected clusters in a random graph. The conjecture is named after its analogy to a bunk bed structure. It was first posited by Pieter Kasteleyn in 1985. [1] A preprint giving a proposed counterexample to the conjecture was posted on the arXiv in October 2024 by Nikita Gladkov, Igor Pak, and Alexander Zimin. [2] [3]
The conjecture has many equivalent formulations. [4] In the most general formulation it involves two identical graphs, referred to as the upper bunk and the lower bunk. These graphs are isomorphic, meaning they share the same structure. Additional edges, termed posts, are added to connect each vertex in the upper bunk with the corresponding vertex in the lower bunk.
Each edge in the graph is assigned a probability. The edges in the upper bunk and their corresponding edges in the lower bunk share the same probability. The probabilities assigned to the posts can be arbitrary.
A random subgraph of the bunkbed graph is then formed by independently deleting each edge based on the assigned probability.
Equivalently, it can be assumed that all edges have the same deletion probability . [4]
The bunkbed conjecture states that in the resulting random subgraph, the probability that a vertex x in the upper bunk is connected to some vertex y in the upper bunk is greater than or equal to the probability that x is connected to y′, the isomorphic copy of y in the lower bunk.
The conjecture suggests that two vertices of a graph are more likely to remain connected after randomly removing some edges if the graph distance between the vertices is smaller. This is intuitive, and similar questions for random walks and Ising model were resolved positively. [5] [6] The original motivation for the conjecture was its implication that, in a percolation on the infinite square grid, the probability of (0, 0) being connected to (x, y) for x, y≥ 0 is greater than the probability of (0, 0) being connected to (x + 1, y). [5]
Despite intuitiveness, proving this conjecture is not straightforward and is an active area of research in percolation theory. [7] It was proved for specific types of graphs, such as wheels, [8] complete graphs, [9] complete bipartite graphs, and graphs with a local symmetry. [10] It was also proved in the limit p→ 1 for any graph. [11] [12] Counterexamples for generalizations of the bunkbed conjecture have been published for site percolation, hypergraphs, and directed graphs. [13]
In combinatorics, Ramsey's theorem, in one of its graph-theoretic forms, states that one will find monochromatic cliques in any edge labelling (with colours) of a sufficiently large complete graph. To demonstrate the theorem for two colours (say, blue and red), let r and s be any two positive integers. Ramsey's theorem states that there exists a least positive integer R(r, s) for which every blue-red edge colouring of the complete graph on R(r, s) vertices contains a blue clique on r vertices or a red clique on s vertices. (Here R(r, s) signifies an integer that depends on both r and s.)
In the mathematical field of graph theory, the Petersen graph is an undirected graph with 10 vertices and 15 edges. It is a small graph that serves as a useful example and counterexample for many problems in graph theory. The Petersen graph is named after Julius Petersen, who in 1898 constructed it to be the smallest bridgeless cubic graph with no three-edge-coloring.
Informally, the reconstruction conjecture in graph theory says that graphs are determined uniquely by their subgraphs. It is due to Kelly and Ulam.
Extremal graph theory is a branch of combinatorics, itself an area of mathematics, that lies at the intersection of extremal combinatorics and graph theory. In essence, extremal graph theory studies how global properties of a graph influence local substructure. Results in extremal graph theory deal with quantitative connections between various graph properties, both global and local, and problems in extremal graph theory can often be formulated as optimization problems: how big or small a parameter of a graph can be, given some constraints that the graph has to satisfy? A graph that is an optimal solution to such an optimization problem is called an extremal graph, and extremal graphs are important objects of study in extremal graph theory.
In the mathematical field of graph theory, an automorphism is a permutation of the vertices such that edges are mapped to edges and non-edges are mapped to non-edges. A graph is a vertex-transitive graph if, given any two vertices v1 and v2 of G, there is an automorphism f such that
In graph theory, a perfect graph is a graph in which the chromatic number equals the size of the maximum clique, both in the graph itself and in every induced subgraph. In all graphs, the chromatic number is greater than or equal to the size of the maximum clique, but they can be far apart. A graph is perfect when these numbers are equal, and remain equal after the deletion of arbitrary subsets of vertices.
In graph theory, the Hadwiger conjecture states that if is loopless and has no minor then its chromatic number satisfies . It is known to be true for . The conjecture is a generalization of the four-color theorem and is considered to be one of the most important and challenging open problems in the field.
In graph theory, a factor of a graph G is a spanning subgraph, i.e., a subgraph that has the same vertex set as G. A k-factor of a graph is a spanning k-regular subgraph, and a k-factorization partitions the edges of the graph into disjoint k-factors. A graph G is said to be k-factorable if it admits a k-factorization. In particular, a 1-factor is a perfect matching, and a 1-factorization of a k-regular graph is a proper edge coloring with k colors. A 2-factor is a collection of cycles that spans all vertices of the graph.
The Zarankiewicz problem, an unsolved problem in mathematics, asks for the largest possible number of edges in a bipartite graph that has a given number of vertices and has no complete bipartite subgraphs of a given size. It belongs to the field of extremal graph theory, a branch of combinatorics, and is named after the Polish mathematician Kazimierz Zarankiewicz, who proposed several special cases of the problem in 1951.
The union-closed sets conjecture, also known as Frankl’s conjecture, is an open problem in combinatorics posed by Péter Frankl in 1979. A family of sets is said to be union-closed if the union of any two sets from the family belongs to the family. The conjecture states:
For every finite union-closed family of sets, other than the family containing only the empty set, there exists an element that belongs to at least half of the sets in the family.
In the mathematical area of graph theory, a triangle-free graph is an undirected graph in which no three vertices form a triangle of edges. Triangle-free graphs may be equivalently defined as graphs with clique number ≤ 2, graphs with girth ≥ 4, graphs with no induced 3-cycle, or locally independent graphs.
In extremal graph theory, the forbidden subgraph problem is the following problem: given a graph , find the maximal number of edges an -vertex graph can have such that it does not have a subgraph isomorphic to . In this context, is called a forbidden subgraph.
In the mathematics of graph drawing, Turán's brick factory problem asks for the minimum number of crossings in a drawing of a complete bipartite graph. The problem is named after Pál Turán, who formulated it while being forced to work in a brick factory during World War II.
In the mathematical field of graph theory, the Horton graph or Horton 96-graph is a 3-regular graph with 96 vertices and 144 edges discovered by Joseph Horton. Published by Bondy and Murty in 1976, it provides a counterexample to the Tutte conjecture that every cubic 3-connected bipartite graph is Hamiltonian.
In graph theory, a k-degenerate graph is an undirected graph in which every subgraph has at least one vertex of degree at most k: that is, some vertex in the subgraph touches k or fewer of the subgraph's edges. The degeneracy of a graph is the smallest value of k for which it is k-degenerate. The degeneracy of a graph is a measure of how sparse it is, and is within a constant factor of other sparsity measures such as the arboricity of a graph.
Barnette's conjecture is an unsolved problem in graph theory, a branch of mathematics, concerning Hamiltonian cycles in graphs. It is named after David W. Barnette, a professor emeritus at the University of California, Davis; it states that every bipartite polyhedral graph with three edges per vertex has a Hamiltonian cycle.
The bipartite realization problem is a classical decision problem in graph theory, a branch of combinatorics. Given two finite sequences and of natural numbers with , the problem asks whether there is a labeled simple bipartite graph such that is the degree sequence of this bipartite graph.
In the mathematical fields of graph theory and finite model theory, the logic of graphs deals with formal specifications of graph properties using sentences of mathematical logic. There are several variations in the types of logical operation that can be used in these sentences. The first-order logic of graphs concerns sentences in which the variables and predicates concern individual vertices and edges of a graph, while monadic second-order graph logic allows quantification over sets of vertices or edges. Logics based on least fixed point operators allow more general predicates over tuples of vertices, but these predicates can only be constructed through fixed-point operators, restricting their power.
In combinatorial mathematics and extremal graph theory, the Ruzsa–Szemerédi problem or (6,3)-problem asks for the maximum number of edges in a graph in which every edge belongs to a unique triangle. Equivalently it asks for the maximum number of edges in a balanced bipartite graph whose edges can be partitioned into a linear number of induced matchings, or the maximum number of triples one can choose from points so that every six points contain at most two triples. The problem is named after Imre Z. Ruzsa and Endre Szemerédi, who first proved that its answer is smaller than by a slowly-growing factor.
Sidorenko's conjecture is a major conjecture in the field of extremal graph theory, posed by Alexander Sidorenko in 1986. Roughly speaking, the conjecture states that for any bipartite graph and graph on vertices with average degree , there are at least labeled copies of in , up to a small error term. Formally, it provides an intuitive inequality about graph homomorphism densities in graphons. The conjectured inequality can be interpreted as a statement that the density of copies of in a graph is asymptotically minimized by a random graph, as one would expect a fraction of possible subgraphs to be a copy of if each edge exists with probability .
