Let T be a hamiltonian bipartite tournament with n vertices, γ a hamiltonian directed cycle of T , and k an even number. In this paper the following question is studied: What is the maximum intersection with γ of a directed cycle of length k contained in T [V (γ)]? It is proved that for an even k in the range n+6 2 ≤ k ≤ n−2, there exists a directed cycle Ch(k) of length h(k), h(k) ∈ {k, k − 2}...

Let V be a finite-dimensional vector space over a field k. A hyperplane arrangement in V is a collection A = (H1, . . . , Hn) of codimension one affine subspaces of V . The arrangement A is called central if the intersection ⋂ Hi is nonempty; without loss of generality the intersection contains the origin. We will always denote by n the number of hyperplanes in the arrangement, and by l the dim...

Given a graph G = (V, E), we define the transpose degree sequence d T j to be equal to the number of vertices of degree at least j. We define L G , the graph Laplacian, to be the matrix, whose rows and columns are indexed by the vertex set V , whose diagonal entry at v is the degree of v and whose value at a pair (v, w) is −1 if (v, w) ∈ E and 0 otherwise. Grone and Merris conjectured [GM] Conj...

We investigate edge-intersection graphs of paths in the plane grid regarding a parameter called the bend-number. The bend-number is related to the interval-number and the track-number of a graph. We provide new upper and lower bounds of the bend-number of any given simple graph in terms of the coloring number, edge clique covers and the maximum degree. We show that the bend-number of an outerpl...

We call a bipartite graph homogeneous if every finite partial automorphism which respects left and right can be extended to a total automorphism. A (κ, λ) bipartite graph is a bipartite graph with left side of size κ and right side of size λ. We show that there is a homogeneous (א0, 2א0) bipartite graph of girth 4 (thus answering negatively a question by Kupitz and Perles), and that depending o...

A bipartite graph G = (L,R;E) with at least one edge is said to be identifiable if for every vertex v ∈ L, the subgraph induced by its non-neighbors has a matching of cardinality |L| − 1. An `-subgraph of G is an induced subgraph of G obtained by deleting from it some vertices in L together with all their neighbors. The Identifiable Subgraph problem is the problem of determining whether a given...

We develop a bipartite rigidity theory for bipartite graphs parallel to the classical rigidity theory for general graphs, and define for two positive integers k, l the notions of (k, l)-rigid and (k, l)-stress free bipartite graphs. This theory coincides with the study of Babson–Novik’s balanced shifting restricted to graphs. We establish bipartite analogs of the cone, contraction, deletion, an...

For suitable subgroups of a finitely generated group, we define the intersection number of one subgroup with another subgroup and show that this number is symmetric. We also give an interpretation of this number. If one considers two simple closed curves L and S on a closed orientable surface F , one can define their intersection number to be the least number of intersection points obtainable b...

Hartman I.B.-A., I. Newman and R. Ziv, On grid intersection graphs, Discrete Mathematics 87 (1991) 41-52. A bipartite graph G = (X, Y; E) has a grid representation if X and Y correspond to sets of horizontal and vertical segments in the plane, respectively, such that (xi, y,) E E if and only if segments x, and y, intersect. We prove that all planar bipartite graphs have a grid representation, a...

A m,n -bipartite graph is a bipartite graph such that one bipartition has m vertices and the other bipartition has n vertices. The tree dumbbell D n, a, b consists of the path Pn−a−b together with a independent vertices adjacent to one pendent vertex of Pn−a−b and b independent vertices adjacent to the other pendent vertex of Pn−a−b. In this paper, firstly, we show that, among m,n bipartite gra...