35,394 research outputs found
On the size-Ramsey number of cycles
For given graphs , the size-Ramsey number is the smallest integer for which there exists a graph on edges such that in every -edge coloring of with colors , contains a monochromatic copy of of color for some . We denote by when .
Haxell, Kohayakawa and \L{}uczak showed that the size-Ramsey number of a cycle is linear in i.e. for some constant . Their proof, however, is based on the regularity lemma of Szemer\'{e}di and so no specific constant is known.
In this paper, we give various upper bounds for the size-Ramsey numbers of cycles. We provide an alternative proof of , avoiding the use of the regularity lemma, where is exponential and doubly-exponential in , when is even and odd, respectively.
In particular, we show that for sufficiently large we have where if is even and otherwise
Ramsey numbers of ordered graphs
An ordered graph is a pair where is a graph and
is a total ordering of its vertices. The ordered Ramsey number
is the minimum number such that every ordered
complete graph with vertices and with edges colored by two colors contains
a monochromatic copy of .
In contrast with the case of unordered graphs, we show that there are
arbitrarily large ordered matchings on vertices for which
is superpolynomial in . This implies that
ordered Ramsey numbers of the same graph can grow superpolynomially in the size
of the graph in one ordering and remain linear in another ordering.
We also prove that the ordered Ramsey number is
polynomial in the number of vertices of if the bandwidth of
is constant or if is an ordered graph of constant
degeneracy and constant interval chromatic number. The first result gives a
positive answer to a question of Conlon, Fox, Lee, and Sudakov.
For a few special classes of ordered paths, stars or matchings, we give
asymptotically tight bounds on their ordered Ramsey numbers. For so-called
monotone cycles we compute their ordered Ramsey numbers exactly. This result
implies exact formulas for geometric Ramsey numbers of cycles introduced by
K\'arolyi, Pach, T\'oth, and Valtr.Comment: 29 pages, 13 figures, to appear in Electronic Journal of
Combinatoric
Bipartite, Size, and Online Ramsey Numbers of Some Cycles and Paths
The basic premise of Ramsey Theory states that in a sufficiently large system, complete disorder is impossible. One instance from the world of graph theory says that given two fixed graphs F and H, there exists a finitely large graph G such that any red/blue edge coloring of the edges of G will produce a red copy of F or a blue copy of H. Much research has been conducted in recent decades on quantifying exactly how large G must be if we consider different classes of graphs for F and H. In this thesis, we explore several Ramsey- type problems with a particular focus on paths and cycles. We first examine the bipartite size Ramsey number of a path on n vertices, bˆr(Pn), and give an upper bound using a random graph construction motivated by prior upper bound improvements in similar problems. Next, we consider the size Ramsey number Rˆ (C, Pn) and provide a significant improvement to the upper bound using a very structured graph, the cube of a path, as opposed to a random construction. We also prove a small improvement to the lower bound and show that the r-colored version of this problem is asymptotically linear in rn. Lastly, we give an upper bound for the online Ramsey number R˜ (C, Pn)
Online size Ramsey numbers: Odd cycles vs connected graphs
Given two graph families and , a size Ramsey
game is played on the edge set of . In every round, Builder
selects an edge and Painter colours it red or blue. Builder is trying to force
Painter to create as soon as possible a red copy of a graph from
or a blue copy of a graph from . The online (size) Ramsey number
is the smallest number of rounds in the
game provided Builder and Painter play optimally. We prove that if is the family of all odd cycles and is the family of all
connected graphs on vertices and edges, then , where is the golden
ratio, and for , we have . We also show that
for . As a consequence we get for every .Comment: 14 pages, 0 figures; added appendix containing intuition behind the
potential function used for lower bound; corrected typos and added a few
clarification
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
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