465 research outputs found
Conic approach to quantum graph parameters using linear optimization over the completely positive semidefinite cone
We investigate the completely positive semidefinite cone ,
a new matrix cone consisting of all matrices that admit a Gram
representation by positive semidefinite matrices (of any size). In particular
we study relationships between this cone and the completely positive and doubly
nonnegative cones, and between its dual cone and trace positive non-commutative
polynomials.
We use this new cone to model quantum analogues of the classical independence
and chromatic graph parameters and , which are roughly
obtained by allowing variables to be positive semidefinite matrices instead of
scalars in the programs defining the classical parameters. We can
formulate these quantum parameters as conic linear programs over the cone
. Using this conic approach we can recover the bounds in
terms of the theta number and define further approximations by exploiting the
link to trace positive polynomials.Comment: Fixed some typo
On topological relaxations of chromatic conjectures
There are several famous unsolved conjectures about the chromatic number that
were relaxed and already proven to hold for the fractional chromatic number. We
discuss similar relaxations for the topological lower bound(s) of the chromatic
number. In particular, we prove that such a relaxed version is true for the
Behzad-Vizing conjecture and also discuss the conjectures of Hedetniemi and of
Hadwiger from this point of view. For the latter, a similar statement was
already proven in an earlier paper of the first author with G. Tardos, our main
concern here is that the so-called odd Hadwiger conjecture looks much more
difficult in this respect. We prove that the statement of the odd Hadwiger
conjecture holds for large enough Kneser graphs and Schrijver graphs of any
fixed chromatic number
Several parameters of generalized Mycielskians
AbstractThe generalized Mycielskians (also known as cones over graphs) are the natural generalization of the Mycielski graphs (which were first introduced by Mycielski in 1955). Given a graph G and any integer m⩾0, one can transform G into a new graph μm(G), the generalized Mycielskian of G. This paper investigates circular clique number, total domination number, open packing number, fractional open packing number, vertex cover number, determinant, spectrum, and biclique partition number of μm(G)
The fractional chromatic number of double cones over graphs
Assume are positive integers and is a graph. Let be the
graph obtained from the path with vertices by adding a loop at vertex . The double cone over a
graph is obtained from the direct product by identifying
into a single vertex , identifying into a single vertex , and adding an edge connecting
and . This paper determines the fractional chromatic
number of . In particular, if or is even, then
, where is the
th cone over . If is odd, then . The chromatic number of is also
discussed.Comment: 23 page
Block-diagonal semidefinite programming hierarchies for 0/1 programming
Lovasz and Schrijver, and later Lasserre, proposed hierarchies of
semidefinite programming relaxations for general 0/1 linear programming
problems. In this paper these two constructions are revisited and two new,
block-diagonal hierarchies are proposed. They have the advantage of being
computationally less costly while being at least as strong as the
Lovasz-Schrijver hierarchy. Our construction is applied to the stable set
problem and experimental results for Paley graphs are reported.Comment: 11 pages, (v2) revision based on suggestions by referee, computation
of N+(TH(P_q)) included in Table
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