3,663 research outputs found
Planar Ramsey numbers for cycles
AbstractFor two given graphs G and H the planar Ramsey number PR(G,H) is the smallest integer n such that every planar graph F on n vertices either contains a copy of G or its complement contains a copy H. By studying the existence of subhamiltonian cycles in complements of sparse graphs, we determine all planar Ramsey numbers for pairs of cycles
Planar Ramsey graphs
We say that a graph is planar unavoidable if there is a planar graph
such that any red/blue coloring of the edges of contains a monochromatic
copy of , otherwise we say that is planar avoidable. I.e., is planar
unavoidable if there is a Ramsey graph for that is planar. It follows from
the Four-Color Theorem and a result of Gon\c{c}alves that if a graph is planar
unavoidable then it is bipartite and outerplanar. We prove that the cycle on
vertices and any path are planar unavoidable. In addition, we prove that
all trees of radius at most are planar unavoidable and there are trees of
radius that are planar avoidable. We also address the planar unavoidable
notion in more than two colors
Spanning embeddings of arrangeable graphs with sublinear bandwidth
The Bandwidth Theorem of B\"ottcher, Schacht and Taraz [Mathematische Annalen
343 (1), 175-205] gives minimum degree conditions for the containment of
spanning graphs H with small bandwidth and bounded maximum degree. We
generalise this result to a-arrangeable graphs H with \Delta(H)<sqrt(n)/log(n),
where n is the number of vertices of H.
Our result implies that sufficiently large n-vertex graphs G with minimum
degree at least (3/4+\gamma)n contain almost all planar graphs on n vertices as
subgraphs. Using techniques developed by Allen, Brightwell and Skokan
[Combinatorica, to appear] we can also apply our methods to show that almost
all planar graphs H have Ramsey number at most 12|H|. We obtain corresponding
results for graphs embeddable on different orientable surfaces.Comment: 20 page
Size-Ramsey numbers of structurally sparse graphs
Size-Ramsey numbers are a central notion in combinatorics and have been
widely studied since their introduction by Erd\H{o}s, Faudree, Rousseau and
Schelp in 1978. Research has mainly focused on the size-Ramsey numbers of
-vertex graphs with constant maximum degree . For example, graphs
which also have constant treewidth are known to have linear size-Ramsey
numbers. On the other extreme, the canonical examples of graphs of unbounded
treewidth are the grid graphs, for which the best known bound has only very
recently been improved from to by Conlon, Nenadov and
Truji\'c. In this paper, we prove a common generalization of these results by
establishing new bounds on the size-Ramsey numbers in terms of treewidth (which
may grow as a function of ). As a special case, this yields a bound of
for proper minor-closed classes of graphs. In
particular, this bound applies to planar graphs, addressing a question of Wood.
Our proof combines methods from structural graph theory and classic
Ramsey-theoretic embedding techniques, taking advantage of the product
structure exhibited by graphs with bounded treewidth.Comment: 21 page
An Extension of Ramsey\u27s Theorem to Multipartite Graphs
Ramsey Theorem, in the most simple form, states that if we are given a positive integer l, there exists a minimal integer r(l), called the Ramsey number, such any partition of the edges of K_r(l) into two sets, i.e. a 2-coloring, yields a copy of K_l contained entirely in one of the partitioned sets, i.e. a monochromatic copy of Kl. We prove an extension of Ramsey\u27s Theorem, in the more general form, by replacing complete graphs by multipartite graphs in both senses, as the partitioned set and as the desired monochromatic graph. More formally, given integers l and k, there exists an integer p(m) such that any 2-coloring of the edges of the complete multipartite graph K_p(m);r(k) yields a monochromatic copy of K_m;k . The tools that are used to prove this result are the Szemeredi Regularity Lemma and the Blow Up Lemma. A full proof of the Regularity Lemma is given. The Blow-Up Lemma is merely stated, but other graph embedding results are given. It is also shown that certain embedding conditions on classes of graphs, namely (f , ?) -embeddability, provides a method to bound the order of the multipartite Ramsey numbers on the graphs. This provides a method to prove that a large class of graphs, including trees, graphs of bounded degree, and planar graphs, has a linear bound, in terms of the number of vertices, on the multipartite Ramsey number
Fixed-Parameter Tractability of Token Jumping on Planar Graphs
Suppose that we are given two independent sets and of a graph
such that , and imagine that a token is placed on each vertex in
. The token jumping problem is to determine whether there exists a
sequence of independent sets which transforms into so that each
independent set in the sequence results from the previous one by moving exactly
one token to another vertex. This problem is known to be PSPACE-complete even
for planar graphs of maximum degree three, and W[1]-hard for general graphs
when parameterized by the number of tokens. In this paper, we present a
fixed-parameter algorithm for the token jumping problem on planar graphs, where
the parameter is only the number of tokens. Furthermore, the algorithm can be
modified so that it finds a shortest sequence for a yes-instance. The same
scheme of the algorithms can be applied to a wider class of graphs,
-free graphs for any fixed integer , and it yields
fixed-parameter algorithms
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