Mathematics

An Euler tour in a hypergraph (also called a rank-2 universal cycle or 1-overlap cycle in the context of designs) is a closed walk that traverses every edge exactly once. In this paper, we define a covering k -hypergraph to be a non-empty k -uniform hypergraph in which every (k??) -subset of vertices appear together in at least one edge. We then show that every covering k -hypergraph, for k?? , admits an Euler tour if and only if it has at least two edges.

We introduce a family of spaces called critical varieties. Each critical variety is a subset of one of positroid varieties in the Grassmannian. The combinatorics of positroid varieties is captured by the dimer model on a planar bipartite graph G , and the critical variety is obtained by restricting to Kenyon's critical dimer model associated to a family of isoradial embeddings of G . This model is invariant under square/spider moves on G , and we give an explicit boundary measurement formula for critical varieties which does not depend on the choice of G . This extends our recent results for the critical Ising model, and simultaneously also includes the case of critical electrical networks. We systematically develop the basic properties of critical varieties. In particular, we study their real and totally positive parts, the combinatorics of the associated strand diagrams, and introduce a shift map motivated by the connection to zonotopal tilings and scattering amplitudes.

We show that 3 -graphs on n vertices whose codegree is at least (2/3+o(1))n can be decomposed into tight cycles and admit Euler tours, subject to the trivial necessary divisibility conditions. We also provide a construction showing that our bounds are best possible up to the o(1) term. All together, our results answer in the negative some recent questions of Glock, Joos, Kühn, and Osthus.

For E??F d q , d?? , where F q is the finite field with q elements, we consider the distance graph G dist t (E) , t?? , where the vertices are the elements of E , and two vertices x , y are connected by an edge if ||x?�y||??( x 1 ??y 1 ) 2 +?? ( x d ??y d ) 2 =t . We prove that if |E|??C k q d+2 2 , then G dist t (E) contains a statistically correct number of cycles of length k . We are also going to consider the dot-product graph G prod t (E) , t?? , where the vertices are the elements of E , and two vertices x , y are connected by an edge if x?�y??x 1 y 1 +?? x d y d =t . We obtain similar results in this case using more sophisticated methods necessitated by the fact that the function x?�y is not translation invariant. The exponent d+2 2 is improved for sufficiently long cycles.

Caccetta-Häggkvist conjecture is a longstanding open problem on degree conditions that force an oriented graph to contain a directed cycle of a bounded length. Motivated by this conjecture, Kelly, Kühn and Osthus initiated a study of degree conditions forcing the containment of a directed cycle of a given length. In particular, they found the optimal minimum semidegree, i.e., the smaller of the minimum indegree and the minimum outdegree, that forces a large oriented graph to contain a directed cycle of a given length not divisible by 3 , and conjectured the optimal minimum semidegree for all the other cycles except the directed triangle. In this paper, we establish the best possible minimum semidegree that forces a large oriented graph to contain a directed cycle of a given length divisible by 3 yet not equal to 3 , hence fully resolve the conjecture of Kelly, Kühn and Osthus. We also find an asymptotically optimal semidegree threshold of any cycle with a given orientation of its edges with the sole exception of a directed triangle.

A signed bipartite (simple) graph (G,?) is said to be C ?? -critical if it admits no homomorphism to C ?? (a negative 4-cycle) but every proper subgraph of it does. In this work, first of all we show that the notion of 4-coloring of graphs and signed graphs is captured, through simple graph operations, by the notion of homomorphism to C ?? . In particular, the 4-color theorem is equivalent to: Given a planar graph G , the signed bipartite graph obtained from G by replacing each edge with a negative path of length 2 maps to C ?? . We prove that, except for one particular signed bipartite graph on 7 vertices and 9 edges, any C ?? -critical signed graph on n vertices must have at least ??4n 3 ??edges, and that this bound or ??4n 3 ??1 is attained for each value of n?? . As an application, we conclude that all signed bipartite planar graphs of negative girth at least 8 map to C ?? . Furthermore, we show that there exists an example of a signed bipartite planar graph of girth 6 which does not map to C ?? , showing 8 is the best possible and disproving a conjecture of Naserasr, Rollova and Sopena, in extension of the above mentioned restatement of the 4CT.

A finite metric space is called here distance degree regular if its distance degree sequence is the same for every vertex. A notion of designs in such spaces is introduced that generalizes that of designs in Q -polynomial distance-regular graphs. An approximation of their cumulative distribution function, based on the notion of Christoffel function in approximation theory is given. As an application we derive limit laws on the weight distributions of binary orthogonal arrays of strength going to infinity. An analogous result for combinatorial designs of strength going to infinity is given.

We give a simple proof of a major index determinant formula in the symmetric group discovered by Krattenthaler and first proved by Thibon using noncommutative symmetric functions. We do so by proving a factorization of an element in the group ring of the symmetric group. By applying similar methods to the groups of signed permutations and colored permutations, we prove determinant formulas in these groups as conjectured by Krattenthaler.

Let Ω n denote the class of n?n doubly stochastic matrices (each such matrix is entrywise nonnegative and every row and column sum is 1). We study the diagonals of matrices in Ω n . The main question is: which A??Ω n are such that the diagonals in A that avoid the zeros of A all have the same sum of their entries. We give a characterization of such matrices, and establish several classes of patterns of such matrices.

Due to their broad application to different fields of theory and practice, generalized Petersen graphs GPG(n,s) have been extensively investigated. Despite the regularity of generalized Petersen graphs, determining an exact formula for the diameter is still a difficult problem. In their paper, Beenker and Van Lint have proved that if the circulant graph C n (1,s) has diameter d , then GPG(n,s) has diameter at least d+1 and at most d+2 . In this paper, we provide necessary and sufficient conditions so that the diameter of GPG(n,s) is equal to d+1, and sufficient conditions so that the diameter of GPG(n,s) is equal to d+2. Afterwards, we give exact values for the diameter of GPG(n,s) for almost all cases of n and s. Furthermore, we show that there exists an algorithm computing the diameter of generalized Petersen graphs with running time O (log n ).