Spider Web Graphs

For given graphs G and H, let |Hom(G, H)| denote the set of graph homomorphisms from G to H. We show that for any finite, n-regular, bipartite graph G and any finite graph H (perhaps with loops), |Hom(G, H)| is maximum when G is a disjoint union of K n,n 's. This generalizes a result of J. Kahn on the number of independent sets in a regular bipartite graph. We also give the asymptotics of the logarithm of |Hom(G, H)| in terms of a simply expressed parameter of H.

formula omitted]-theory of locally finite graph [formula omitted]-algebras

Axiomatic characterization of the interval function of a bipartite graph

In this paper, we prove the recently proposed family of bipartite graphs, Spider Web Networks [9], are hamiltonian laceable. More precisely, let G=(V, E) be a spider web network with bipartitions C and D. For any pair of nodes {x, y| x∈C, y∈D}, there is a path P between x and y where P visits all nodes in V exactly once.

Following this advice, we should look over the big questions, then doodle and sketch out some approaches. If you are an expert, then this is easy to do, but most people do not want to wait to become an expert before looking at interesting problems. Graph theory, our area of expertise, has many hard-to-solve questions. Some hark back to the recreational roots of the area yet still keep their mystery. These ‘‘acorns’’ can be planted on the backs of envelopes, on a blackboard, and over a coffee. Our goal is to collect some of these conjectures—arguably some of the most intriguing—in one place. We present ten conjectures in graph theory, and you can read about each one in at most ten minutes. As we live in the era of Twitter, all the conjectures we state are 140 characters or fewer (so ‘‘minute’’ here has a double meaning). We might even call these sketchy tweets, as we present examples for each conjecture that you can doodle on as you read. Hamming also references ambiguity: good researchers can work both on proving and disproving the same statement, so we approach the conjectures with an open mind. He also mentions that a good approach is to reframe the problem, and change the point of view. One example from Vizing’s Conjecture (which is discussed as our second-to-last conjecture in the following text), is the three-page paper [2] which, with a new way of thinking, reduced most of the published work of twenty years to a corollary of its main result! Given the size of modern graph theory, with its many smaller subfields (such as structural graph theory, random graphs, topological graph theory, graph algorithms, spectral graph theory, graph minors, and graph homomorphisms, to name a few), it would be impossible to list all, or even the bulk of the conjectures in the field. We are content instead to focus on a few family jewels, which have an intrinsic beauty and have provided some challenges for graph-theorists for at least two decades. There is something for everyone here, from undergraduate students taking their first course in graph theory, to seasoned researchers in the field. For additional reading on problems and conjectures in graph theory and other fields, see the Open Problem Garden maintained by IRMACS at Simon Fraser University [24]. We consider only finite and undirected graphs, with no multiple edges or loops (unless otherwise stated). We assume the reader has some basic familiarity with graphs and their terminology, including notions such as cycles, paths, complete graphs, complete bipartite graphs, vertex degrees, and connected graphs. We use the notation Cn for the cycle with n vertices, Pn for the path with n vertices, and Kn for the complete graph with n vertices. The complete bipartite graph with m and n vertices of the respective colors is denoted by Km,n. All the background we need can be found in any text in graph theory, such as those of Diestel [9] and West [41], or online (see for example [42]).

Use is made of an auxiliary matrix to count trees of complete bipartite graphs. One form of the auxiliary matrix achieves partial diagonalization of the node admittance matrix and a second form of the same matrix completes the enumeration. Extension of the results is shown to be compatible with a known count for trees of complete graphs.

Families of Induced Trees and Their Intersection Graphs

Constructing trees in bipartite graphs

Homomorphisms of infinite bipartite graphs onto complete bipartite graphs

Cycles through all finite vertex sets in infinite graphs

A bipartite graph is a system consisting of two sets of vertices S and T and a set of edges K, each edge joining a vertex of S to a vertex of T. A set U of edges of K is said to be independent if no two edges of U have a vertex in common. The largest possible number of independent edges has been variously called the exterior dimension [3], term rank [4, 5, 7], etc. This number is the same as the smallest number of vertices in a set W such that each edge of K has at least one of its vertices in W. The edges of a finite bipartite graph can be represented as a set of cells in a matrix as follows. If S = a1, a2, …, an T = b1, b2, … bm, the edges of K are represented by some of the cells of an n by m matrix as follows: if K contains the edge joining ai to bj then the (i, j)th cell of the matrix represents this edge. It is convenient sometimes to represent the set K by a matrix A with real entries aij where aij = 0 if ai is not joined to bj in K and aij > 0 if ai is joined to bj in K. Any non-null graph K will have infinitely many matrix representations.

Constraint satisfaction problems (CSPs) are one of the central topics in theoretical computer science, in particular, in the area of artificial intelligence. Their computational complexity is due to relatively recent results from areas of mathematics, including finite-model-theory, algebra and graph homomorphisms. The main conjecture by Feder and Vardi states that any CSP over a finite relational template is either in P or is NP-complete. Further, it amounts to showing that every non NP-complete CSP can be expressed as an extension of first-order logic. A finite template is Mal'tsev, a compatible algebraic operation, which is closely related to an affine space over a finite field. The so-called Bulatov-Dalmau algorithm, a natural generalization of the Gaussian elimination on vector spaces, shows such CSPs are tractable. In this work, we prove that CSPs described over a finite template Mal'tsev are expressible in logic LFP+rnk, providing a logical proof that such CSPs are tractable.

A defect set in a bipartite graph with vertex classes V and W is a subset X ⊂ V such that the neighbourhood N(X) satisfies |N(X)| < |X|. We study a lemma on defect sets in bipartite graphs with certain expanding properties from the algorithmic complexity point of view. This lemma is the core of a result of Friedman and Pippenger which states that expanding graphs contain all small trees. We also discuss related problems of finding shortest circuits of matroids represented over a field. In particular, we propose a new straightforward method to derive a weaker form (PR-completeness) of the recent NP-completeness results of Khachiyan [11] and Vardy [18] concerning this problem for the field of rationals and GF(p m), respectively.KeywordsPolynomial TimeDecision ProblemBipartite GraphShort CircuitFinite FieldThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In this thesis we study the number of spanning trees in some classes of graphs. This is made possible by the famous matrix tree theorem established by Kirchhoff in 1847 which states that the number of spanning trees in a finite graph is given by the product of the non-zero eigenvalues of the combinatorial Laplacian of the graph divided by the number of vertices. We adapt techniques derived by Chinta, Jorgenson and Karlsson in 2010 for d-dimensional discrete tori to circulant graphs with first generator equals to 1 and to d-dimensional degenerating discrete tori. They are degenerating in the sense that d-p sides of the tori are tending to infinity at the same rate while the p other sides tend to infinity sublinearly with respect to the d-p sides. Furthermore, the results on d-dimensional discrete tori enable to derive asymptotics for the number of spanning trees on d-dimensional orthotope square lattices. Other results obtained in this thesis concern closed formulas for the number of spanning trees in directed and non-directed circulant graphs where the generators vary, that is, they linearly depend on the number of vertices.

Recent results establish for 2-spin antiferromagnetic systems that the computational complexity of approximating the partition function on graphs of maximum degree D undergoes a phase transition that coincides with the uniqueness phase transition on the infinite D-regular tree. For the ferromagnetic Potts model we investigate whether analogous hardness results hold. Goldberg and Jerrum showed that approximating the partition function of the ferromagnetic Potts model is at least as hard as approximating the number of independent sets in bipartite graphs (#BIS-hardness). We improve this hardness result by establishing it for bipartite graphs of maximum degree D. We first present a detailed picture for the phase diagram for the infinite D-regular tree, giving a refined picture of its first-order phase transition and establishing the critical temperature for the coexistence of the disordered and ordered phases. We then prove for all temperatures below this critical temperature that it is #BIS-hard to approximate the partition function on bipartite graphs of maximum degree D. As a corollary, it is #BIS-hard to approximate the number of k-colorings on bipartite graphs of maximum degree D when k <= D/(2 ln D). The #BIS-hardness result for the ferromagnetic Potts model uses random bipartite regular graphs as a gadget in the reduction. The analysis of these random graphs relies on recent connections between the maxima of the expectation of their partition function, attractive fixpoints of the associated tree recursions, and induced matrix norms. We extend these connections to random regular graphs for all ferromagnetic models and establish the Bethe prediction for every ferromagnetic spin system on random regular graphs. We also prove for the ferromagnetic Potts model that the Swendsen-Wang algorithm is torpidly mixing on random D-regular graphs at the critical temperature for large q.