Slovenija Abstract A partial extension of the results in 1], where 2-arc-transitive cir-culants are classiied, is given. It is proved that a 2-arc-transitive Cayley graph of a dihedral group is either a complete graph or a bi-partite graph.

Let G = (V;E) be a t-partite graph with n vertices and m edges, where the partite sets are given. We present an O(n2m1:5) time algorithm to construct drawings of G in the plane so that the size of the largest set of pairwise crossing edges, and at the same time, the size of the largest set of disjoint (pairwise non-crossing) edges are O(pt m). As an application we embed G in a book of O(pt m) p...

We present a novel approach for robust localization of multiple people observed using a set of static cameras. We use this location information to generate a visualization of the virtual offside line in soccer games. To compute the position of the offside line, we need to localize players′ positions, and identify their team roles. We solve the problem of fusing corresponding players′ positional...

We investigate separability of Laplacian matrices of graphs when seen as density matrices. This is a family of quantum states with many combinatorial properties. We firstly show that the well-known matrix realignment criterion can be used to test separability of this type of quantum states. The criterion can be interpreted as novel graph-theoretic idea. Then, we prove that the density matrix of...

An edge dominating set in a graph G is a set of edges D such that every edge not in D is adjacent to an edge of D. An edge domatic partition of a graph C=(V, E) is a collection of pairwise-disjoint edge dominating sets of G whose union is E. The maximum size of an edge domatic partition of G is called the edge domatic number. In this paper, we study the edge domatic number of the complete parti...

We introduce a search problem for finding a regular bi-partite graph of maximum attainable girth for specified degree and number of vertices, by restricting the search space using a series of mathematically rigourous arguments from [1] and [2]. The goal of this paper is to derive the enumeration search algorithm for finding a girth maximum (m, r) BTU, which is notation for regular partite graph...

For a graph F and a function f : N → R, let ef (F ) = ∑ x∈V (F ) f(d(x)) and let exf (n, F ) be the maximum of ef (G) over all F -free graphs G with n vertices. Suppose that f is a non-decreasing function with the property that for any ε > 0 there is δ > 0 such that for any n ≤ m ≤ (1+δ)n we have f(m) ≤ (1+ε)f(n). Under this assumption we prove that the asymptotics of exf (n, F ), where F is a ...

Polar graphs are a natural extension of some classes of graphs like bipartite graphs, split graphs and complements of bipartite graphs. A graph is (s, k)-polar if there exists a partition A, B of its vertex set such that A induces a complete s-partite graph (i.e., a collection of at most s disjoint stable sets with complete links between all sets) and B a disjoint union of at most k cliques (i....

Let G be a bipartite graph, and let λe, λi be two parallel convex curves; we study the question about whether G admits a planar straight line drawing such that the vertices of one partite set of G lie on λe and the vertices of the other partite set lie on λi. A characterization is presented that gives rise to linear time testing and drawing algorithms.

In this paper we extend a classical theorem of Corrádi and Hajnal into the setting of sparse random graphs. We show that if p(n) (log n/n), then asymptotically almost surely every subgraph of G(n, p) with minimum degree at least (2/3 + o(1))np contains a triangle packing that covers all but at most O(p−2) vertices. Moreover, the assumption on p is optimal up to the (log n) factor and the presen...