2,096 research outputs found
On the multi-colored ramsey numbers of paths and even cycles
In this paper we improve the upper bound on the multi-color Ramsey numbers of paths and even cycles. More precisely, we prove the following. For every r ≥ 2 there exists an n0 = n0(r) such that for n ≥ n0 we have (Formula Presented). For every r ≥ 2 and even n we have (Formula Presented). The main tool is a stability version of the Erdős-Gallai theorem that may be of independent interest. © 2016 Australian National University. All rights reserved
Improved bounds on the multicolor Ramsey numbers of paths and even cycles
We study the multicolor Ramsey numbers for paths and even cycles,
and , which are the smallest integers such that every coloring of
the complete graph has a monochromatic copy of or
respectively. For a long time, has only been known to lie between
and . A recent breakthrough by S\'ark\"ozy and later
improvement by Davies, Jenssen and Roberts give an upper bound of . We improve the upper bound to . Our approach uses structural insights in connected graphs without a
large matching. These insights may be of independent interest
On globally sparse Ramsey graphs
We say that a graph has the Ramsey property w.r.t.\ some graph and
some integer , or is -Ramsey for short, if any -coloring
of the edges of contains a monochromatic copy of . R{\"o}dl and
Ruci{\'n}ski asked how globally sparse -Ramsey graphs can possibly
be, where the density of is measured by the subgraph with
the highest average degree. So far, this so-called Ramsey density is known only
for cliques and some trivial graphs . In this work we determine the Ramsey
density up to some small error terms for several cases when is a complete
bipartite graph, a cycle or a path, and colors are available
On small Mixed Pattern Ramsey numbers
We call the minimum order of any complete graph so that for any coloring of
the edges by colors it is impossible to avoid a monochromatic or rainbow
triangle, a Mixed Ramsey number. For any graph with edges colored from the
above set of colors, if we consider the condition of excluding in the
above definition, we produce a \emph{Mixed Pattern Ramsey number}, denoted
. We determine this function in terms of for all colored -cycles
and all colored -cliques. We also find bounds for when is a
monochromatic odd cycles, or a star for sufficiently large . We state
several open questions.Comment: 16 page
Monochromatic connected matchings in 2-edge-colored multipartite graphs
A matching in a graph is connected if all the edges of are in the
same component of . Following \L uczak,there have been many results using
the existence of large connected matchings in cluster graphs with respect to
regular partitions of large graphs to show the existence of long paths and
other structures in these graphs. We prove exact
Ramsey-type bounds on the sizes of monochromatic connected matchings in
-edge-colored multipartite graphs. In addition, we prove a stability theorem
for such matchings.Comment: 29 pages, 2 figure
Approximation bounds on maximum edge 2-coloring of dense graphs
For a graph and integer , an edge -coloring of is an
assignment of colors to edges of , such that edges incident on a vertex span
at most distinct colors. The maximum edge -coloring problem seeks to
maximize the number of colors in an edge -coloring of a graph . The
problem has been studied in combinatorics in the context of {\em anti-Ramsey}
numbers. Algorithmically, the problem is NP-Hard for and assuming the
unique games conjecture, it cannot be approximated in polynomial time to a
factor less than . The case , is particularly relevant in practice,
and has been well studied from the view point of approximation algorithms. A
-factor algorithm is known for general graphs, and recently a -factor
approximation bound was shown for graphs with perfect matching. The algorithm
(which we refer to as the matching based algorithm) is as follows: "Find a
maximum matching of . Give distinct colors to the edges of . Let
be the connected components that results when M is
removed from G. To all edges of give the th color."
In this paper, we first show that the approximation guarantee of the matching
based algorithm is for graphs with perfect matching
and minimum degree . For , this is better than the approximation guarantee proved in {AAAP}. For triangle free graphs
with perfect matching, we prove that the approximation factor is , which is better than for .Comment: 11pages, 3 figure
Monochromatic loose paths in multicolored -uniform cliques
For integers and , a -uniform hypergraph is called a
loose path of length , and denoted by , if it consists of
edges such that if and
if . In other words, each pair of
consecutive edges intersects on a single vertex, while all other pairs are
disjoint. Let be the minimum integer such that every
-edge-coloring of the complete -uniform hypergraph yields a
monochromatic copy of . In this paper we are mostly interested in
constructive upper bounds on , meaning that on the cost of
possibly enlarging the order of the complete hypergraph, we would like to
efficiently find a monochromatic copy of in every coloring. In
particular, we show that there is a constant such that for all ,
, , and , there is an
algorithm such that for every -edge-coloring of the edges of , it
finds a monochromatic copy of in time at most . We also
prove a non-constructive upper bound
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