9,782 research outputs found
Relations between the local chromatic number and its directed version
The local chromatic number is a coloring parameter defined as the minimum number of colors that should appear in the most colorful closed neighborhood of a vertex under any proper coloring of the graph. Its directed version is the same when we consider only outneighborhoods in a directed graph. For digraphs with all arcs being present in both directions the two values are obviously equal. Here, we consider oriented graphs. We show the existence of a graph where the directed local chromatic number of all oriented versions of the graph is strictly less than the local chromatic number of the underlying undirected graph. We show that for fractional versions the analogous problem has a different answer: there always exists an orientation for which the directed and undirected values coincide. We also determine the supremum of the possible ratios of these fractional parameters, which turns out to be e, the basis of the natural logarithm
Graph Theory versus Minimum Rank for Index Coding
We obtain novel index coding schemes and show that they provably outperform
all previously known graph theoretic bounds proposed so far. Further, we
establish a rather strong negative result: all known graph theoretic bounds are
within a logarithmic factor from the chromatic number. This is in striking
contrast to minrank since prior work has shown that it can outperform the
chromatic number by a polynomial factor in some cases. The conclusion is that
all known graph theoretic bounds are not much stronger than the chromatic
number.Comment: 8 pages, 2 figures. Submitted to ISIT 201
Hipergráfok = Hypergraphs
A projekt cĂ©lkitűzĂ©seit sikerĂĽlt megvalĂłsĂtani. A nĂ©gy Ă©v során több mint száz kiválĂł eredmĂ©ny szĂĽletett, amibĹ‘l eddig 84 dolgozat jelent meg a tĂ©ma legkiválĂłbb folyĂłirataiban, mint Combinatorica, Journal of Combinatorial Theory, Journal of Graph Theory, Random Graphs and Structures, stb. Számos rĂ©gĂłta fennállĂł sejtĂ©st bebizonyĂtottunk, egĂ©sz rĂ©gi nyitott problĂ©mát megoldottunk hipergráfokkal kapcsolatban illetve kapcsolĂłdĂł terĂĽleteken. A problĂ©mák nĂ©melyike sok Ă©ve, olykor több Ă©vtizede nyitott volt. Nem egy közvetlen kutatási eredmĂ©ny, de szintĂ©n bizonyos Ă©rtĂ©kmĂ©rĹ‘, hogy a rĂ©sztvevĹ‘k egyike a NorvĂ©g Királyi AkadĂ©mia tagja lett Ă©s elnyerte a Steele dĂjat. | We managed to reach the goals of the project. We achieved more than one hundred excellent results, 84 of them appeared already in the most prestigious journals of the subject, like Combinatorica, Journal of Combinatorial Theory, Journal of Graph Theory, Random Graphs and Structures, etc. We proved several long standing conjectures, solved quite old open problems in the area of hypergraphs and related subjects. Some of the problems were open for many years, sometimes for decades. It is not a direct research result but kind of an evaluation too that a member of the team became a member of the Norvegian Royal Academy and won Steele Prize
Quasi-Parallel Segments and Characterization of Unique Bichromatic Matchings
Given n red and n blue points in general position in the plane, it is
well-known that there is a perfect matching formed by non-crossing line
segments. We characterize the bichromatic point sets which admit exactly one
non-crossing matching. We give several geometric descriptions of such sets, and
find an O(nlogn) algorithm that checks whether a given bichromatic set has this
property.Comment: 31 pages, 24 figure
Structure and enumeration of (3+1)-free posets
A poset is (3+1)-free if it does not contain the disjoint union of chains of
length 3 and 1 as an induced subposet. These posets play a central role in the
(3+1)-free conjecture of Stanley and Stembridge. Lewis and Zhang have
enumerated (3+1)-free posets in the graded case by decomposing them into
bipartite graphs, but until now the general enumeration problem has remained
open. We give a finer decomposition into bipartite graphs which applies to all
(3+1)-free posets and obtain generating functions which count (3+1)-free posets
with labelled or unlabelled vertices. Using this decomposition, we obtain a
decomposition of the automorphism group and asymptotics for the number of
(3+1)-free posets.Comment: 28 pages, 5 figures. New version includes substantial changes to
clarify the construction of skeleta and the enumeration. An extended abstract
of this paper appears as arXiv:1212.535
Complete Acyclic Colorings
We study two parameters that arise from the dichromatic number and the
vertex-arboricity in the same way that the achromatic number comes from the
chromatic number. The adichromatic number of a digraph is the largest number of
colors its vertices can be colored with such that every color induces an
acyclic subdigraph but merging any two colors yields a monochromatic directed
cycle. Similarly, the a-vertex arboricity of an undirected graph is the largest
number of colors that can be used such that every color induces a forest but
merging any two yields a monochromatic cycle. We study the relation between
these parameters and their behavior with respect to other classical parameters
such as degeneracy and most importantly feedback vertex sets.Comment: 17 pages, no figure
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