25 research outputs found
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
Nonrepetitive Colouring via Entropy Compression
A vertex colouring of a graph is \emph{nonrepetitive} if there is no path
whose first half receives the same sequence of colours as the second half. A
graph is nonrepetitively -choosable if given lists of at least colours
at each vertex, there is a nonrepetitive colouring such that each vertex is
coloured from its own list. It is known that every graph with maximum degree
is -choosable, for some constant . We prove this result
with (ignoring lower order terms). We then prove that every subdivision
of a graph with sufficiently many division vertices per edge is nonrepetitively
5-choosable. The proofs of both these results are based on the Moser-Tardos
entropy-compression method, and a recent extension by Grytczuk, Kozik and Micek
for the nonrepetitive choosability of paths. Finally, we prove that every graph
with pathwidth is nonrepetitively -colourable.Comment: v4: Minor changes made following helpful comments by the referee
On a conjecture of Mąkowski and Schinzel concerning the composition of the arithmetic functions σ and ϕ
For any positive integer n let ϕ(n) and σ(n) be the Euler function of n and the sum of divisors of n, respectively. In [5], Mąkowski and Schinzel conjectured that the inequality σ(ϕ(n)) ≥ n/2 holds for all positive integers n. We show that the lower density of the set of positive integers satisfying the above inequality is at least 0.74
On-line Ramsey Theory
The Ramsey game we consider in this paper is played on an unbounded set of vertices by two players, called Builder and Painter. In one move Builder introduces a new edge and Painter paints it red or blue. The goal of Builder is to force Painter to create a monochromatic copy of a fixed target graph H, keeping the constructed graph in a prescribed class G. The main problem is to recognize the winner for a given pair H, G. In particular, we prove that Builder has a winning strategy for any k-colorable graph H in the game played on k-colorable graphs. Another class of graphs with this strange self-unavoidability property is the class of forests. We show that the class of outerplanar graphs does not have this property. The question of whether planar graphs are self-unavoidable is left open. We also consider a multicolor version of Ramsey on-line game. To extend our main result for 3-colorable graphs we introduce another Ramsey type game, which seems interesting in its own right.
The game of arboricity
AbstractUsing a fixed set of colors C, Ann and Ben color the edges of a graph G so that no monochromatic cycle may appear. Ann wins if all edges of G have been colored, while Ben wins if completing a coloring is not possible. The minimum size of C for which Ann has a winning strategy is called the game arboricity of G, denoted by Ag(G). We prove that Ag(G)⩽3k for any graph G of arboricity k, and that there are graphs such that Ag(G)⩾2k-2. The upper bound is achieved by a suitable version of the activation strategy, used earlier for the vertex coloring game. We also provide two other strategies based on induction and acyclic colorings