5 research outputs found
Parallel cleaning of a network with brushes
AbstractWe consider the process of cleaning a network where at each time step, all vertices that have at least as many brushes as incident, contaminated edges, send brushes down these edges and remove them from the network. An added condition is that, because of the contamination model used, the final configuration must be the initial configuration of another cleaning of the network. We find the minimum number of brushes required for trees, cycles, complete bipartite networks; and for all networks when all edges must be cleaned on each step. Finally, we give bounds on the number of brushes required for complete networks
Game Brush Number
We study a two-person game based on the well-studied brushing process on
graphs. Players Min and Max alternately place brushes on the vertices of a
graph. When a vertex accumulates at least as many brushes as its degree, it
sends one brush to each neighbor and is removed from the graph; this may in
turn induce the removal of other vertices. The game ends once all vertices have
been removed. Min seeks to minimize the number of brushes played during the
game, while Max seeks to maximize it. When both players play optimally, the
length of the game is the game brush number of the graph , denoted .
By considering strategies for both players and modelling the evolution of the
game with differential equations, we provide an asymptotic value for the game
brush number of the complete graph; namely, we show that . Using a fractional version of the game, we couple the game
brush numbers of complete graphs and the binomial random graph
. It is shown that for asymptotically almost
surely .
Finally, we study the relationship between the game brush number and the
(original) brush number.Comment: 20 pages, 3 figure