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

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...

On Graphs Which Contain All Sparse Graphs

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...

Vertices cannot be hidden from quantum spatial search for almost all random graphs

Adam Glos, 2 Ryszard Kukulski, 3 Aleksandra Krawiec, 3, ∗ and Zbigniew Pucha la 4 Institute of Theoretical and Applied Informatics, Polish Academy of Sciences, ul. Ba ltycka 5, 44-100 Gliwice, Poland Institute of Informatics, Silesian University of Technology, ul. Akademicka 16, 44-100 Gliwice, Poland Institute of Mathematics, University of Silesia, ul. Bankowa 14, 40-007 Katowice, Poland Facul...

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...