2,495 research outputs found
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, bËr(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 RË (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 RË (C, Pn)
Applications of a new separator theorem for string graphs
An intersection graph of curves in the plane is called a string graph.
Matousek almost completely settled a conjecture of the authors by showing that
every string graph of m edges admits a vertex separator of size O(\sqrt{m}\log
m). In the present note, this bound is combined with a result of the authors,
according to which every dense string graph contains a large complete balanced
bipartite graph. Three applications are given concerning string graphs G with n
vertices: (i) if K_t is not a subgraph of G for some t, then the chromatic
number of G is at most (\log n)^{O(\log t)}; (ii) if K_{t,t} is not a subgraph
of G, then G has at most t(\log t)^{O(1)}n edges,; and (iii) a lopsided
Ramsey-type result, which shows that the Erdos-Hajnal conjecture almost holds
for string graphs.Comment: 7 page
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
Connected matchings in special families of graphs.
A connected matching in a graph is a set of disjoint edges such that, for any pair of these edges, there is another edge of the graph incident to both of them. This dissertation investigates two problems related to finding large connected matchings in graphs. The first problem is motivated by a famous and still open conjecture made by Hadwiger stating that every k-chromatic graph contains a minor of the complete graph Kk . If true, Hadwiger\u27s conjecture would imply that every graph G has a minor of the complete graph K n/a(C), where a(G) denotes the independence number of G. For a graph G with a(G) = 2, ThomassĂ© first noted the connection between connected matchings and large complete graph minors: there exists an ? \u3e 0 such that every graph G with a( G) = 2 contains K ?+, as a minor if and only if there exists a positive constant c such that every graph G with a( G) = 2 contains a connected matching of size cn. In Chapter 3 we prove several structural properties of a vertexminimal counterexample to these statements, extending work by Blasiak. We also prove the existence of large connected matchings in graphs with clique size close to the Ramsey bound by proving: for any positive constants band c with c \u3c ÂŒ, there exists a positive integer N such that, if G is a graph with n =: N vertices, 0\u27( G) = 2, and clique size at most bv(n log(n) )then G contains a connected matching of size cn. The second problem concerns computational complexity of finding the size of a maximum connected matching in a graph. This problem has many applications including, when the underlying graph is chordal bipartite, applications to the bipartite margin shop problem. For general graphs, this problem is NP-complete. Cameron has shown the problem is polynomial-time solvable for chordal graphs. Inspired by this and applications to the margin shop problem, in Chapter 4 we focus on the class of chordal bipartite graphs and one of its subclasses, the convex bipartite graphs. We show that a polynomial-time algorithm to find the size of a maximum connected matching in a chordal bipartite graph reduces to finding a polynomial-time algorithm to recognize chordal bipartite graphs that have a perfect connected matching. We also prove that, in chordal bipartite graphs, a connected matching of size k is equivalent to several other statements about the graph and its biadjacency matrix, including for example, the statement that the complement of the latter contains a k x k submatrix that is permutation equivalent to strictly upper triangular matrix
Applications of a New Separator Theorem for String Graphs
An intersection graph of curves in the plane is called a string graph. MatouĆĄek almost completely settled a conjecture of the authors by showing that every string graph with m edges admits a vertex separator of size . In the present note, this bound is combined with a result of the authors, according to which every dense string graph contains a large complete balanced bipartite graph. Three applications are given concerning string graphs G with n vertices: (i) if Kt â G for some t, then the chromatic number of G is at most (log n) O(log t); (ii) if Kt,t â G, then G has at most t(log t) O(1) n edges,; and (iii) a lopsided Ramsey-type result, which shows that the ErdĆs-Hajnal conjecture almost holds for string graph
Upper density problems in infinite Ramsey theory
We consider the following question in infinite Ramsey theory, introduced by ErdĆs and Galvin [EG93] in a particular case and by DeBiasio and McKenney [DM19] in a more general setting. Let H be a countably infinite graph. If the edges of the complete graph on the natural numbers are colored red or blue, what is the maximum value of λ such that we are guaranteed to find a monochromatic copy of H whose vertex set has upper density at least λ? We call this value the Ramsey density of H.
The problem of determining the Ramsey density of the infinite path was first studied by ErdĆs and Galvin, and was recently solved by Corsten, DeBiasio, Lang and the author [CDLL19]. In this thesis we study the problem of determining the Ramsey density of arbitrary graphs H. On an intuitive level, we show that three properties of a graph H have an effect on the Ramsey density: the chromatic number, the number of components, and the expansion of its independent sets. We deduce the exact value of the Ramsey density for a wide variety of graphs, including all locally finite forests, bipartite factors, clique factors and odd cycle factors. We also determine the value of the Ramsey density of all locally finite graphs, up to a factor of 2.
We also study a list coloring variant of the same problem. We show that there exists a way of assigning a list of size two to every edge in the complete graph on N such that, in every list coloring, there are monochromatic paths with density arbitrarily close to 1.Wir betrachten die folgende Fragestellung aus der Ramsey-Theorie, welche von ErdĆs und Galvin [EG93] in einem Spezialfall sowie von DeBiasio und McKenney [DM19] in einem allgemeineren Kontext formuliert wurde: Es sei H ein abzĂ€hlbar unendlicher Graph. Welches ist der gröĂtmögliche Wert λ, sodass wir, wenn die Kanten des vollstĂ€ndigen Graphen mit Knotenmenge N jeweils entweder rot oder blau gefĂ€rbt sind, stets eine einfarbige Kopie von H, dessen Knotenmenge eine obere asymptotische Dichte von mindestens λ besitzt, finden können? Wir nennen diesen Wert die Ramsey-Dichte von H.
Das Problem, die Ramsey-Dichte des unendlichen Pfades zu bestimmen wurde erstmals von ErdĆs und Galvin untersucht und wurde vor kurzem von Corsten, DeBiasio, Lang und dem Autor [CDLL19] gelöst. Gegenstand der vorliegenden Dissertation ist die Bestimmung der Ramsey-Dichten von Graphen. Auf einer intuitiven Ebene zeigen wir, dass drei Parameter eines Graphen die Ramsey-Dichte beeinflussen: die chromatische Zahl, die Anzahl der Zusammenhangskomponenten sowie die Expansion seiner unabhĂ€ngigen Mengen. Wir ermitteln die exakten Werte der Ramsey-Dichte fĂŒr eine Vielzahl von Graphen, darunter alle lokal endlichen WĂ€lder, bipartite Faktoren, Kr-Faktoren sowie Ck-Faktoren fĂŒr ungerade k. Ferner bestimmen wir den Wert der Ramsey-Dichte aller lokal endlichen Graphen bis auf einen Faktor 2.
DarĂŒber hinaus untersuchen wir eine Variante des oben beschriebenen Problems fĂŒr ListenfĂ€rbungen. Wir zeigen, dass es möglich ist, jeder Kante des vollstĂ€ndigen Graphen mit Knotenmenge N eine Liste der GröĂe Zwei zuzuweisen, sodass in jeder zugehörigen ListenfĂ€rbung monochromatische Pfade mit beliebig nah an 1 liegender Dichte existieren
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
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
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