10 research outputs found
Walker-Breaker Games on
The Maker-Breaker connectivity game and Hamilton cycle game belong to the
best studied games in positional games theory, including results on biased
games, games on random graphs and fast winning strategies. Recently, the
Connector-Breaker game variant, in which Connector has to claim edges such that
her graph stays connected throughout the game, as well as the Walker-Breaker
game variant, in which Walker has to claim her edges according to a walk, have
received growing attention.
For instance, London and Pluh\'ar studied the threshold bias for the
Connector-Breaker connectivity game on a complete graph , and showed that
there is a big difference between the cases when Maker's bias equals or
. Moreover, a recent result by the first and third author as well as Kirsch
shows that the threshold probability for the Connector-Breaker
connectivity game on a random graph is of order
. We extent this result further to Walker-Breaker games and
prove that this probability is also enough for Walker to create a Hamilton
cycle
-cross -intersecting families via necessary intersection points
Given integers and we call families
-cross
-intersecting if for all , , we have
. We obtain a strong generalisation of
the classic Hilton-Milner theorem on cross intersecting families. In
particular, we determine the maximum of for -cross -intersecting families in the
cases when these are -uniform families or arbitrary subfamilies of
. Only some special cases of these results had been proved
before. We obtain the aforementioned theorems as instances of a more general
result that considers measures of -cross -intersecting families. This
also provides the maximum of for
families of possibly mixed uniformities .Comment: 13 page
Random perturbation of sparse graphs
In the model of randomly perturbed graphs we consider the union of a deterministic graph Gα with minimum degree αn and the binomial random graph G(n, p). This model was introduced by Bohman, Frieze, and Martin and for Hamilton cycles their result bridges the gap between Diracâs theorem and the results by PĂłsa and Korshunov on the threshold in G(n, p). In this note we extend this result in Gα âȘG(n, p) to sparser graphs with α = o(1). More precisely, for any Δ > 0 and α: N âŠâ (0, 1) we show that a.a.s. Gα âȘ G(n, ÎČ/n) is Hamiltonian, where ÎČ = â(6 + Δ) log(α). If α > 0 is a fixed constant this gives the aforementioned result by Bohman, Frieze, and Martin and if α = O(1/n) the random part G(n, p) is sufficient for a Hamilton cycle. We also discuss embeddings of bounded degree trees and other spanning structures in this model, which lead to interesting questions on almost spanning embeddings into G(n, p)
Fast Strategies in Waiter-Client Games on
Waiter-Client games are played on some hypergraph , where
denotes the family of winning sets. For some bias , during
each round of such a game Waiter offers to Client elements of , of
which Client claims one for himself while the rest go to Waiter. Proceeding
like this Waiter wins the game if she forces Client to claim all the elements
of any winning set from . In this paper we study fast strategies
for several Waiter-Client games played on the edge set of the complete graph,
i.e. , in which the winning sets are perfect matchings, Hamilton
cycles, pancyclic graphs, fixed spanning trees or factors of a given graph.Comment: 38 page
Waiter-Client games on randomly perturbed graphs
Waiter-Client games are played on a hypergraph (X, F), where Fâ 2X denotes the family of winning sets. During each round, Waiter offers a predefined amount (called bias) of elements from the board X, from which Client takes one for himself while the rest go to Waiter. Waiter wins the game if she can force Client to occupy any winning set Fâ F. In this paper we consider Waiter-Client games played on randomly perturbed graphs. These graphs consist of the union of a deterministic graph Gα on n vertices with minimum degree at least αn and the binomial random graph Gn , p. Depending on the bias we determine the order of the threshold probability for winning the Hamiltonicity game and the k-connectivity game on GαâȘ Gn , p