1,511 research outputs found
Constrained Ramsey Numbers
For two graphs S and T, the constrained Ramsey number f(S, T) is the minimum
n such that every edge coloring of the complete graph on n vertices, with any
number of colors, has a monochromatic subgraph isomorphic to S or a rainbow
(all edges differently colored) subgraph isomorphic to T. The Erdos-Rado
Canonical Ramsey Theorem implies that f(S, T) exists if and only if S is a star
or T is acyclic, and much work has been done to determine the rate of growth of
f(S, T) for various types of parameters. When S and T are both trees having s
and t edges respectively, Jamison, Jiang, and Ling showed that f(S, T) <=
O(st^2) and conjectured that it is always at most O(st). They also mentioned
that one of the most interesting open special cases is when T is a path. In
this work, we study this case and show that f(S, P_t) = O(st log t), which
differs only by a logarithmic factor from the conjecture. This substantially
improves the previous bounds for most values of s and t.Comment: 12 pages; minor revision
Generalizations of the Tree Packing Conjecture
The Gy\'arf\'as tree packing conjecture asserts that any set of trees with
vertices has an (edge-disjoint) packing into the complete graph
on vertices. Gy\'arf\'as and Lehel proved that the conjecture holds in some
special cases. We address the problem of packing trees into -chromatic
graphs. In particular, we prove that if all but three of the trees are stars
then they have a packing into any -chromatic graph. We also consider several
other generalizations of the conjecture
Red-blue clique partitions and (1-1)-transversals
Motivated by the problem of Gallai on -transversals of -intervals,
it was proved by the authors in 1969 that if the edges of a complete graph
are colored with red and blue (both colors can appear on an edge) so that there
is no monochromatic induced and then the vertices of can be
partitioned into a red and a blue clique. Aharoni, Berger, Chudnovsky and Ziani
recently strengthened this by showing that it is enough to assume that there is
no induced monochromatic and there is no induced in {\em one of the
colors}. Here this is strengthened further, it is enough to assume that there
is no monochromatic induced and there is no on which both color
classes induce a .
We also answer a question of Kaiser and Rabinovich, giving an example of six
-convex sets in the plane such that any three intersect but there is no
-transversal for them
Vertex covers by monochromatic pieces - A survey of results and problems
This survey is devoted to problems and results concerning covering the
vertices of edge colored graphs or hypergraphs with monochromatic paths, cycles
and other objects. It is an expanded version of the talk with the same title at
the Seventh Cracow Conference on Graph Theory, held in Rytro in September
14-19, 2014.Comment: Discrete Mathematics, 201
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