1,148 research outputs found
Game chromatic number of Cartesian and corona product graphs
The game chromatic number is investigated for Cartesian product and corona product of two graphs and . The exact values for the game chromatic number of Cartesian product graph of is found, where is a star graph of order . This extends previous results of Bartnicki et al. [1] and Sia [5] on the game chromatic number of Cartesian product graphs. Let be the path graph on vertices and be the cycle graph on vertices. We have determined the exact values for the game chromatic number of corona product graphs and
Kochen-Specker Sets and the Rank-1 Quantum Chromatic Number
The quantum chromatic number of a graph is sandwiched between its
chromatic number and its clique number, which are well known NP-hard
quantities. We restrict our attention to the rank-1 quantum chromatic number
, which upper bounds the quantum chromatic number, but is
defined under stronger constraints. We study its relation with the chromatic
number and the minimum dimension of orthogonal representations
. It is known that . We
answer three open questions about these relations: we give a necessary and
sufficient condition to have , we exhibit a class of
graphs such that , and we give a necessary and
sufficient condition to have . Our main tools are
Kochen-Specker sets, collections of vectors with a traditionally important role
in the study of noncontextuality of physical theories, and more recently in the
quantification of quantum zero-error capacities. Finally, as a corollary of our
results and a result by Avis, Hasegawa, Kikuchi, and Sasaki on the quantum
chromatic number, we give a family of Kochen-Specker sets of growing dimension.Comment: 12 page
Sabidussi Versus Hedetniemi for Three Variations of the Chromatic Number
We investigate vector chromatic number, Lovasz theta of the complement, and
quantum chromatic number from the perspective of graph homomorphisms. We prove
an analog of Sabidussi's theorem for each of these parameters, i.e. that for
each of the parameters, the value on the Cartesian product of graphs is equal
to the maximum of the values on the factors. We also prove an analog of
Hedetniemi's conjecture for Lovasz theta of the complement, i.e. that its value
on the categorical product of graphs is equal to the minimum of its values on
the factors. We conjecture that the analogous results hold for vector and
quantum chromatic number, and we prove that this is the case for some special
classes of graphs.Comment: 18 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
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