80 research outputs found
Distribution of colors in Gallai colorings
A Gallai coloring is an edge coloring that avoids triangles colored with
three different colors. Given integers with
for some , does there exist a Gallai
-coloring of with edges in color ? In this paper, we give
several sufficient conditions and one necessary condition to guarantee a
positive answer to the above question. In particular, we prove the existence of
a Gallai-coloring if and . We prove
that for any integer there is a (unique) integer with the
following property: there exists a Gallai -coloring of with
edges in color for every satisfying
, if and only if . We show that
, , and for every
Transitive and Gallai colorings
A Gallai coloring of the complete graph is an edge-coloring with no rainbow
triangle. This concept first appeared in the study of comparability graphs and
anti-Ramsey theory. We introduce a transitive analogue for acyclic directed
graphs, and generalize both notions to Coxeter systems, matroids and
commutative algebras.
It is shown that for any finite matroid (or oriented matroid), the maximal
number of colors is equal to the matroid rank. This generalizes a result of
Erd\H{o}s-Simonovits-S\'os for complete graphs. The number of Gallai (or
transitive) colorings of the matroid that use at most colors is a
polynomial in . Also, for any acyclic oriented matroid, represented over the
real numbers, the number of transitive colorings using at most 2 colors is
equal to the number of chambers in the dual hyperplane arrangement.
We count Gallai and transitive colorings of the root system of type A using
the maximal number of colors, and show that, when equipped with a natural
descent set map, the resulting quasisymmetric function is symmetric and
Schur-positive.Comment: 31 pages, 5 figure
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
Ramsey-nice families of graphs
For a finite family of fixed graphs let be
the smallest integer for which every -coloring of the edges of the
complete graph yields a monochromatic copy of some . We
say that is -nice if for every graph with
and for every -coloring of there exists a
monochromatic copy of some . It is easy to see that if
contains no forest, then it is not -nice for any . It seems
plausible to conjecture that a (weak) converse holds, namely, for any finite
family of graphs that contains at least one forest, and for all
(or at least for infinitely many values of ),
is -nice. We prove several (modest) results in support of this
conjecture, showing, in particular, that it holds for each of the three
families consisting of two connected graphs with 3 edges each and observing
that it holds for any family containing a forest with at most 2
edges. We also study some related problems and disprove a conjecture by
Aharoni, Charbit and Howard regarding the size of matchings in regular
3-partite 3-uniform hypergraphs.Comment: 20 pages, 2 figure
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