A graph which embeds all small graphs on any large set of vertices

A graph which embeds all small graphs on any large set of vertices

The main objective of this paper will be to construct a graph on o1 which has the property that every finite graph appears as an induced subgraph in any uncountable set of vertices. Similar questions have been considered by Erdiis and Hajnal [l, 21. The construction to be presented was motivated by 0 3 of [3] which follows Todor&viPs proof that c&[o& in [6]. The same result was also obtained in...

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

The random graph embeds in the curve graph of any infinite genus surface

The random graph is an infinite graph with the universal property that any embedding of G−v extends to an embedding of G, for any finite graph. In this paper we show that this graph embeds in the curve graph of a surface Σ if and only if Σ has infinite genus, showing that the curve system on an infinite genus surface is “as complicated as possible”.

Occupation games on graphs in which the second player takes almost all vertices

Given a connected graph G = (V,E), two players take turns occupying vertices v ∈ V by putting black and white tokens so that the current vertex sets B,W ⊆ V are disjoint, B ∩W = ∅, and the corresponding induced subgraphs G[B] and G[W ] are connected any time. A player must pass whenever (s)he has no legal move. (Obviously, after this, the opponent will take all remaining vertices, since G is as...