1,143 research outputs found
An algorithm to prescribe the configuration of a finite graph
We provide algorithms involving edge slides, for a connected simple graph to
evolve in a finite number of steps to another connected simple graph in a
prescribed configuration, and for the regularization of such a graph by the
minimization of an appropriate energy functional
Strong Ramsey Games in Unbounded Time
For two graphs and the strong Ramsey game on the
board and with target is played as follows. Two players alternately
claim edges of . The first player to build a copy of wins. If none of
the players win, the game is declared a draw. A notorious open question of Beck
asks whether the first player has a winning strategy in
in bounded time as . Surprisingly, in a recent paper Hefetz
et al. constructed a -uniform hypergraph for which they proved
that the first player does not have a winning strategy in
in bounded time. They naturally ask
whether the same result holds for graphs. In this paper we make further
progress in decreasing the rank.
In our first result, we construct a graph (in fact )
and prove that the first player does not have a winning strategy in
in bounded time. As an application of this
result we deduce our second result in which we construct a -uniform
hypergraph and prove that the first player does not have a winning
strategy in in bounded time. This improves the
result in the paper above.
An equivalent formulation of our first result is that the game
is a draw. Another reason for interest
on the board is a folklore result that the disjoint
union of two finite positional games both of which are first player wins is
also a first player win. An amusing corollary of our first result is that at
least one of the following two natural statements is false: (1) for every graph
, is a first player win; (2) for every graph
if is a first player win, then
is also a first player win.Comment: 17 pages, 48 figures; improved presentation, particularly in section
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