67 research outputs found
Decomposition of bounded degree graphs into -free subgraphs
We prove that every graph with maximum degree admits a partition of
its edges into parts (as ) none of which
contains as a subgraph. This bound is sharp up to a constant factor. Our
proof uses an iterated random colouring procedure.Comment: 8 pages; to appear in European Journal of Combinatoric
THE ELECTRONIC JOURNAL OF COMBINATORICS (2014), DS1.14 References
and Computing 11. The results of 143 references depend on computer algorithms. The references are ordered alphabetically by the last name of the first author, and where multiple papers have the same first author they are ordered by the last name of the second author, etc. We preferred that all work by the same author be in consecutive positions. Unfortunately, this causes that some of the abbreviations are not in alphabetical order. For example, [BaRT] is earlier on the list than [BaLS]. We also wish to explain a possible confusion with respect to the order of parts and spelling of Chinese names. We put them without any abbreviations, often with the last name written first as is customary in original. Sometimes this is different from the citations in other sources. One can obtain all variations of writing any specific name by consulting the authors database of Mathematical Reviews a
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
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