Graphs which Contain all Small Graphs

On graphs which contain all small trees

Communicated by the Managing Editors We investigate those graphs G, with the property that any tree on N vertices occurs as subgraph of G,. In particular, we consider the problem of estimating the minimum number of edges such a graph can have. We show that this number is bounded below and above by $z log II and nl+l/log log %, respectively. A typical question in extremal graph theory1 is one wh...

On Graphs Which Contain All Sparse Graphs

Almost all cop-win graphs contain a universal vertex

We consider cop-win graphs in the binomial random graph G(n, 1/2). We prove that almost all cop-win graphs contain a universal vertex. From this result, we derive that the asymptotic number of labelled cop-win graphs of order n is equal to (1 + o(1))n2 2/2−3n/2+1.

All Ramsey (2K2,C4)−Minimal Graphs

Let F, G and H be non-empty graphs. The notation F → (G,H) means that if any edge of F is colored by red or blue, then either the red subgraph of F con- tains a graph G or the blue subgraph of F contains a graph H. A graph F (without isolated vertices) is called a Ramsey (G,H)−minimal if F → (G,H) and for every e ∈ E(F), (F − e) 9 (G,H). The set of all Ramsey (G,H)−minimal graphs is denoted by ...

All Good Drawings of Small Complete Graphs ∗

Good drawings (also known as simple topological graphs) are drawings of graphs such that any two edges intersect at most once. Such drawings have attracted attention as generalizations of geometric graphs, in connection with the crossing number, and as data structures in their own right. We are in particular interested in good drawings of the complete graph. In this extended abstract, we descri...