477 research outputs found
k-tuple colorings of the Cartesian product of graphs
A k-tuple coloring of a graph G assigns a set of k colors to each vertex of G such that if two vertices are adjacent, the corresponding sets of colors are disjoint. The k-tuple chromatic number of G, χk(G), is the smallest t so that there is a k-tuple coloring of G using t colors. It is well known that χ(Gâ–¡H)=max{χ(G),χ(H)}. In this paper, we show that there exist graphs G and H such that χk(Gâ–¡H)>max{χk(G),χk(H)} for k≥2. Moreover, we also show that there exist graph families such that, for any k≥1, the k-tuple chromatic number of their Cartesian product is equal to the maximum k-tuple chromatic number of its factors.Fil: Bonomo, Flavia. Consejo Nacional de Investigaciones CientÃficas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigación en Ciencias de la Computación. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigación en Ciencias de la Computación; Argentina. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Computación; ArgentinaFil: Koch, Ivo Valerio. Universidad Nacional de General Sarmiento. Instituto de Industria; ArgentinaFil: Torres, Pablo. Universidad Nacional de Rosario. Facultad de Ciencias Exactas, IngenierÃa y Agrimensura; Argentina. Consejo Nacional de Investigaciones CientÃficas y Técnicas. Centro CientÃfico Tecnológico Conicet - Rosario; ArgentinaFil: Valencia Pabon, Mario. Universite de Paris 13-Nord; Francia. Centre National de la Recherche Scientifique; Franci
Structural Properties of Index Coding Capacity Using Fractional Graph Theory
The capacity region of the index coding problem is characterized through the
notion of confusion graph and its fractional chromatic number. Based on this
multiletter characterization, several structural properties of the capacity
region are established, some of which are already noted by Tahmasbi, Shahrasbi,
and Gohari, but proved here with simple and more direct graph-theoretic
arguments. In particular, the capacity region of a given index coding problem
is shown to be simple functionals of the capacity regions of smaller
subproblems when the interaction between the subproblems is none, one-way, or
complete.Comment: 5 pages, to appear in the 2015 IEEE International Symposium on
Information Theory (ISIT
List Distinguishing Parameters of Trees
A coloring of the vertices of a graph G is said to be distinguishing}
provided no nontrivial automorphism of G preserves all of the vertex colors.
The distinguishing number of G, D(G), is the minimum number of colors in a
distinguishing coloring of G. The distinguishing chromatic number of G,
chi_D(G), is the minimum number of colors in a distinguishing coloring of G
that is also a proper coloring.
Recently the notion of a distinguishing coloring was extended to that of a
list distinguishing coloring. Given an assignment L= {L(v) : v in V(G)} of
lists of available colors to the vertices of G, we say that G is (properly)
L-distinguishable if there is a (proper) distinguishing coloring f of G such
that f(v) is in L(v) for all v. The list distinguishing number of G, D_l(G), is
the minimum integer k such that G is L-distinguishable for any list assignment
L with |L(v)| = k for all v. Similarly, the list distinguishing chromatic
number of G, denoted chi_{D_l}(G) is the minimum integer k such that G is
properly L-distinguishable for any list assignment L with |L(v)| = k for all v.
In this paper, we study these distinguishing parameters for trees, and in
particular extend an enumerative technique of Cheng to show that for any tree
T, D_l(T) = D(T), chi_D(T)=chi_{D_l}(T), and chi_D(T) <= D(T) + 1.Comment: 10 page
A Generalization of Kochen-Specker Sets Relates Quantum Coloring to Entanglement-Assisted Channel Capacity
We introduce two generalizations of Kochen-Specker (KS) sets: projective KS
sets and generalized KS sets. We then use projective KS sets to characterize
all graphs for which the chromatic number is strictly larger than the quantum
chromatic number. Here, the quantum chromatic number is defined via a nonlocal
game based on graph coloring. We further show that from any graph with
separation between these two quantities, one can construct a classical channel
for which entanglement assistance increases the one-shot zero-error capacity.
As an example, we exhibit a new family of classical channels with an
exponential increase.Comment: 16 page
Hamilton paths with lasting separation
We determine the asymptotics of the largest cardinality of a set of Hamilton
paths in the complete graph with vertex set [n] under the condition that for
any two of the paths in the family there is a subpath of length k entirely
contained in only one of them and edge{disjoint from the other one
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