64 research outputs found
Bounded colorings of multipartite graphs and hypergraphs
Let be an edge-coloring of the complete -vertex graph . The
problem of finding properly colored and rainbow Hamilton cycles in was
initiated in 1976 by Bollob\'as and Erd\H os and has been extensively studied
since then. Recently it was extended to the hypergraph setting by Dudek, Frieze
and Ruci\'nski. We generalize these results, giving sufficient local (resp.
global) restrictions on the colorings which guarantee a properly colored (resp.
rainbow) copy of a given hypergraph .
We also study multipartite analogues of these questions. We give (up to a
constant factor) optimal sufficient conditions for a coloring of the
complete balanced -partite graph to contain a properly colored or rainbow
copy of a given graph with maximum degree . Our bounds exhibit a
surprising transition in the rate of growth, showing that the problem is
fundamentally different in the regimes and Our
main tool is the framework of Lu and Sz\'ekely for the space of random
bijections, which we extend to product spaces
A note on balanced edge-colorings avoiding rainbow cliques of size four
A balanced edge-coloring of the complete graph is an edge-coloring such that
every vertex is incident to each color the same number of times. In this short
note, we present a construction of a balanced edge-coloring with six colors of
the complete graph on vertices, for every positive integer , with
no rainbow . This solves a problem by Erd\H{o}s and Tuza.Comment: 2 page
Rainbow Generalizations of Ramsey Theory - A Dynamic Survey
In this work, we collect Ramsey-type results concerning rainbow edge colorings of graphs
Rainbow Generalizations of Ramsey Theory - A Dynamic Survey
In this work, we collect Ramsey-type results concerning rainbow edge colorings of graphs
Rainbow Generalizations of Ramsey Theory - A Dynamic Survey
In this work, we collect Ramsey-type results concerning rainbow edge colorings of graphs
Rainbow Subgraphs in Edge-colored Complete Graphs -- Answering two Questions by Erd\H{o}s and Tuza
An edge-coloring of a complete graph with a set of colors is called
completely balanced if any vertex is incident to the same number of edges of
each color from . Erd\H{o}s and Tuza asked in whether for any graph
on edges and any completely balanced coloring of any sufficiently
large complete graph using colors contains a rainbow copy of . This
question was restated by Erd\H{o}s in his list of ``Some of my favourite
problems on cycles and colourings''. We answer this question in the negative
for most cliques by giving explicit constructions of respective
completely balanced colorings. Further, we answer a related question concerning
completely balanced colorings of complete graphs with more colors than the
number of edges in the graph .Comment: 8 page
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
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