John Haslegrave

An evasion game on a graph

Abstract This paper introduces a pursuit and evasion game to be played on a connected graph. One player moves invisibly around the graph, and the other player must guess his position. At each time step the second player guesses a vertex, winning if it is the current location of the first player; if not the first player must move along an edge. It is shown that the graphs on which the second player can guarantee to win are precisely the trees that do not contain a particular forbidden subgraph, and best possible capture times on such graphs are obtained.

Judicious partitions of uniform hypergraphs

AbstractThe vertices of any graph with m edges may be partitioned into two parts so that each part meets at least

Extremal results on average subtree density of series-reduced trees

\tfrac{{2m}} {3}

Preferential attachment with choice

edges. Bollobás and Thomason conjectured that the vertices of any r-uniform hypergraph with m edges may likewise be partitioned into r classes such that each part meets at least

Subdivisions in the Robber Locating game

\tfrac{r} {{2r - 1}}

Reaching consensus on a connected graph

edges. In this paper we prove the weaker statement that, for each r ≥ 4, a partition into r classes may be found in which each class meets at least