6 research outputs found
On-line Ramsey numbers
Consider the following game between two players, Builder and Painter. Builder
draws edges one at a time and Painter colours them, in either red or blue, as
each appears. Builder's aim is to force Painter to draw a monochromatic copy of
a fixed graph G. The minimum number of edges which Builder must draw,
regardless of Painter's strategy, in order to guarantee that this happens is
known as the on-line Ramsey number \tilde{r}(G) of G. Our main result, relating
to the conjecture that \tilde{r}(K_t) = o(\binom{r(t)}{2}), is that there
exists a constant c > 1 such that \tilde{r}(K_t) \leq c^{-t} \binom{r(t)}{2}
for infinitely many values of t. We also prove a more specific upper bound for
this number, showing that there exists a constant c such that \tilde{r}(K_t)
\leq t^{-c \frac{\log t}{\log \log t}} 4^t. Finally, we prove a new upper bound
for the on-line Ramsey number of the complete bipartite graph K_{t,t}.Comment: 11 page
On-line Ramsey numbers of paths and cycles
Consider a game played on the edge set of the infinite clique by two players,
Builder and Painter. In each round, Builder chooses an edge and Painter colours
it red or blue. Builder wins by creating either a red copy of or a blue
copy of for some fixed graphs and . The minimum number of rounds
within which Builder can win, assuming both players play perfectly, is the
on-line Ramsey number . In this paper, we consider the case
where is a path . We prove that for all , and determine
) up to an additive constant for all .
We also prove some general lower bounds for on-line Ramsey numbers of the form
.Comment: Preprin
Online Ramsey theory for a triangle on -free graphs
Given a class of graphs and a fixed graph , the online
Ramsey game for on is a game between two players Builder and
Painter as follows: an unbounded set of vertices is given as an initial state,
and on each turn Builder introduces a new edge with the constraint that the
resulting graph must be in , and Painter colors the new edge either
red or blue. Builder wins the game if Painter is forced to make a monochromatic
copy of at some point in the game. Otherwise, Painter can avoid creating a
monochromatic copy of forever, and we say Painter wins the game.
We initiate the study of characterizing the graphs such that for a given
graph , Painter wins the online Ramsey game for on -free graphs. We
characterize all graphs such that Painter wins the online Ramsey game for
on the class of -free graphs, except when is one particular graph.
We also show that Painter wins the online Ramsey game for on the class of
-minor-free graphs, extending a result by Grytczuk, Ha{\l}uszczak, and
Kierstead.Comment: 20 pages, 10 page
Packings and coverings with Hamilton cycles and on-line Ramsey theory
A major theme in modern graph theory is the exploration of maximal packings and minimal covers of graphs with subgraphs in some given family. We focus on packings and coverings with Hamilton cycles, and prove the following results in the area.
• Let ε > 0, and let be a large graph on n vertices with minimum degree at least (1=2+ ε)n. We give a tight lower bound on the size of a maximal packing of with edge-disjoint Hamilton cycles.
• Let be a strongly k-connected tournament. We give an almost tight lower bound on the size of a maximal packing of with edge-disjoint Hamilton cycles.
• Let log /≤≤1-. We prove that may a.a.s be covered by a set of ⌈Δ()/2⌉ Hamilton cycles, which is clearly best possible.
In addition, we consider some problems in on-line Ramsey theory. Let r(,) denote the on-line Ramsey number of and . We conjecture the exact values of r (,) for all ≤ℓ. We prove this conjecture for =2, prove it to within an additive error of 10 for =3, and prove an asymptotically tight lower bound for =4. We also determine r(, exactly for all ℓ