7,340 research outputs found
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
Refined activation strategy for the marking game
AbstractThis paper introduces a new strategy for playing the marking game on graphs. Using this strategy, we prove that if G is a planar graph, then the game colouring number of G, and hence the game chromatic number of G, is at most 17
Bears with Hats and Independence Polynomials
Consider the following hat guessing game. A bear sits on each vertex of a
graph , and a demon puts on each bear a hat colored by one of colors.
Each bear sees only the hat colors of his neighbors. Based on this information
only, each bear has to guess colors and he guesses correctly if his hat
color is included in his guesses. The bears win if at least one bear guesses
correctly for any hat arrangement.
We introduce a new parameter - fractional hat chromatic number ,
arising from the hat guessing game. The parameter is related to the
hat chromatic number which has been studied before. We present a surprising
connection between the hat guessing game and the independence polynomial of
graphs. This connection allows us to compute the fractional hat chromatic
number of chordal graphs in polynomial time, to bound fractional hat chromatic
number by a function of maximum degree of , and to compute the exact value
of of cliques, paths, and cycles
The Eternal Game Chromatic Number of a Graph
Game coloring is a well-studied two-player game in which each player properly
colors one vertex of a graph at a time until all the vertices are colored. An
`eternal' version of game coloring is introduced in this paper in which the
vertices are colored and re-colored from a color set over a sequence of rounds.
In a given round, each vertex is colored, or re-colored, once, so that a proper
coloring is maintained. Player 1 wants to maintain a proper coloring forever,
while player 2 wants to force the coloring process to fail. The eternal game
chromatic number of a graph is defined to be the minimum number of colors
needed in the color set so that player 1 can always win the game on . We
consider several variations of this new game and show its behavior on some
elementary classes of graphs
A new graph based on the semi-direct product of some monoids
In this paper, firstly, we define a new graph based on the semi-direct product of a free abelian monoid of rank n by a finite cyclic monoid, and then discuss some graph properties on this new graph, namely diameter, maximum and minimum degrees, girth, degree sequence and irregularity index, domination number, chromatic number, clique number of (PM). Since graph theoretical studies (including such above graph parameters) consist of some fixed point techniques, they have been applied in fields such as chemistry (in the meaning of atoms, molecules, energy etc.) and engineering (in the meaning of signal processing etc.), game theory and physics
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