Let G be a graph without loops or multiple edges drawn in the plane. It is shown that, for any k, if G has at least Ckn edges and n vertices, then it contains three sets of k edges, such that every edge in any of the sets crosses all edges in the other two sets. Furthermore, two of the three sets can be chosen such that all k edges in the set have a common vertex.

We derive an upper bound on the number of vertices in graphs of diameter 3 and given degree arising from Abelian lifts of dipoles with loops and multiple edges.

. We think that if φ(e) = uv (short for {u, v}), then e connects u and v or equivalently e is incident to u and v. If there is at least one edge incident to u and v then we say that the vertices u and v are adjacent. If φ(e) = vv, then we call e a loop at v. If several edges e1, . . . , ej (j > 1) satisfy φ(e1) = · · · = φ(ej) = uv, then we say that there is a multiple edge between u and v. A g...

The growing popularity and diversity of social network applications present new opportunities as well as new challenges. The resulting social networks have high value to business intelligence, sociological studies, organizational studies, epidemical studies, etc. The ability to explore and extract information of interest from the networks is thus crucial. However, these networks are often large...

We study clustering on graphs with multiple edge types. Our main motivation is that similarities between objects can be measured in many different metrics. For instance similarity between two papers can be based on common authors, where they are published, keyword similarity, citations, etc. As such, graphs with multiple edges is a more accurate model to describe similarities between objects. E...

This paper deals with weighted set systems (V, E , q), where V is a set of indices, E ⊂ 2 and the weight q is a nonnegative integer function on E . The basic idea of the paper is to apply weighted set systems to formulate restrictions on intersections. It is of interest to know whether a weighted set system can be represented by set intersections. An intersection representation of (V, E , q) is...

A pseudo-triangulation on a closed surface without loops is a graph embedded on the surface so that each face is triangular and may have multiple edges, but no lo $0$ps. We shall establish a theory of diagonal flips in those pseudotriangulations. Our theory will work in parallel to that for simple triangulations basically, but it will present more concrete theorems than the latter.

A universal cycle is a compact listing of a class of combinatorial objects. In this paper, we prove the existence of universal cycles of classes of labeled graphs, including simple graphs, trees, graphs with m edges, graphs with loops, graphs with multiple edges (with up to m duplications of each edge), directed graphs, hypergraphs, and k-uniform hypergraphs.

The graphs we shall consider are topological graphs that is they lie in R3 and each edge is homeomorphic to [0, 1]. If a graph is simple, that is it has no loops or multiple edges, then each edge may be taken to be a straight line joining the two vertices at the ends of the edge. A function is k-to-1 if each point in the codomain has precisely k preimages in the domain. Given two graphs G and H...

A k-tree is a chordal graph with no (k + 2)-clique. An l-treepartition of a graph G is a vertex partition of G into ‘bags’, such that contracting each bag to a single vertex gives an l-tree (after deleting loops and replacing parallel edges by a single edge). We prove that for all k ≥ l ≥ 0, every k-tree has an l-tree-partition in which every bag induces a connected ⌊k/(l + 1)⌋-tree. An analogo...