1,389 research outputs found
Hadwiger Number and the Cartesian Product Of Graphs
The Hadwiger number mr(G) of a graph G is the largest integer n for which the
complete graph K_n on n vertices is a minor of G. Hadwiger conjectured that for
every graph G, mr(G) >= chi(G), where chi(G) is the chromatic number of G. In
this paper, we study the Hadwiger number of the Cartesian product G [] H of
graphs.
As the main result of this paper, we prove that mr(G_1 [] G_2) >= h\sqrt{l}(1
- o(1)) for any two graphs G_1 and G_2 with mr(G_1) = h and mr(G_2) = l. We
show that the above lower bound is asymptotically best possible. This
asymptotically settles a question of Z. Miller (1978).
As consequences of our main result, we show the following:
1. Let G be a connected graph. Let the (unique) prime factorization of G be
given by G_1 [] G_2 [] ... [] G_k. Then G satisfies Hadwiger's conjecture if k
>= 2.log(log(chi(G))) + c', where c' is a constant. This improves the
2.log(chi(G))+3 bound of Chandran and Sivadasan.
2. Let G_1 and G_2 be two graphs such that chi(G_1) >= chi(G_2) >=
c.log^{1.5}(chi(G_1)), where c is a constant. Then G_1 [] G_2 satisfies
Hadwiger's conjecture.
3. Hadwiger's conjecture is true for G^d (Cartesian product of G taken d
times) for every graph G and every d >= 2. This settles a question by Chandran
and Sivadasan (They had shown that the Hadiwger's conjecture is true for G^d if
d >= 3.)Comment: 10 pages, 2 figures, major update: lower and upper bound proofs have
been revised. The bounds are now asymptotically tigh
Conic Optimization Theory: Convexification Techniques and Numerical Algorithms
Optimization is at the core of control theory and appears in several areas of
this field, such as optimal control, distributed control, system
identification, robust control, state estimation, model predictive control and
dynamic programming. The recent advances in various topics of modern
optimization have also been revamping the area of machine learning. Motivated
by the crucial role of optimization theory in the design, analysis, control and
operation of real-world systems, this tutorial paper offers a detailed overview
of some major advances in this area, namely conic optimization and its emerging
applications. First, we discuss the importance of conic optimization in
different areas. Then, we explain seminal results on the design of hierarchies
of convex relaxations for a wide range of nonconvex problems. Finally, we study
different numerical algorithms for large-scale conic optimization problems.Comment: 18 page
Factoring cardinal product graphs in polynomial time
AbstractIn this paper a polynomial algorithm for the prime factorization of finite, connected nonbipartite graphs with respect to the cardinal product is presented. This algorithm also decomposes finite, connected graphs into their prime factors with respect to the strong product and provides the basis for a new proof of the uniqueness of the prime factorization of finite, connected nonbipartite graphs with respect to the cardinal product. Furthermore, some of the consequences of these results and several open problems are discussed
Fastest mixing Markov chain on graphs with symmetries
We show how to exploit symmetries of a graph to efficiently compute the
fastest mixing Markov chain on the graph (i.e., find the transition
probabilities on the edges to minimize the second-largest eigenvalue modulus of
the transition probability matrix). Exploiting symmetry can lead to significant
reduction in both the number of variables and the size of matrices in the
corresponding semidefinite program, thus enable numerical solution of
large-scale instances that are otherwise computationally infeasible. We obtain
analytic or semi-analytic results for particular classes of graphs, such as
edge-transitive and distance-transitive graphs. We describe two general
approaches for symmetry exploitation, based on orbit theory and
block-diagonalization, respectively. We also establish the connection between
these two approaches.Comment: 39 pages, 15 figure
- …