2,820 research outputs found
Graph classes with linear Ramsey numbers
The Ramsey number for a class of graphs is the minimum such that every graph in with at least vertices has either a clique of size or an independent set of size . We say that Ramsey numbers are linear in if there is a constant such that for all . In the present paper we conjecture that if is a hereditary class defined by finitely many forbidden induced subgraphs, then Ramsey numbers are linear in if and only if excludes a forest, a disjoint union of cliques and their complements. We prove the "only if" part of this conjecture and verify the "if" part for a variety of classes. We also apply the notion of linearity to bipartite Ramsey numbers and reveal a number of similarities and differences between the bipartite and non-bipartite case
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, bΛr(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 RΛ (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 RΛ (C, Pn)
On Size Multipartite Ramsey Numbers for Stars Versus Paths and Cycles
Let be a complete, balanced, multipartite graph consisting of partite sets and vertices in each partite set. For given two graphs and , and integer , the size multipartite Ramsey number is the smallest integer such that every factorization of the graph satisfies the following condition: either contains or contains . In 2007, Syafrizal et al. determined the size multipartite Ramsey numbers of paths versus stars, for only. Furthermore, Surahmat et al. (2014) gave the size tripartite Ramsey numbers of paths versus stars, for . In this paper, we investigate the size tripartite Ramsey numbers of paths versus stars, with all . Our results complete the previous results given by Syafrizal et al. and Surahmat et al. We also determine the size bipartite Ramsey numbers of stars versus cycles, for
Density theorems for bipartite graphs and related Ramsey-type results
In this paper, we present several density-type theorems which show how to
find a copy of a sparse bipartite graph in a graph of positive density. Our
results imply several new bounds for classical problems in graph Ramsey theory
and improve and generalize earlier results of various researchers. The proofs
combine probabilistic arguments with some combinatorial ideas. In addition,
these techniques can be used to study properties of graphs with a forbidden
induced subgraph, edge intersection patterns in topological graphs, and to
obtain several other Ramsey-type statements
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
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