Lower bounds for superpatterns and universal sequences

Superpatterns and Universal Point Sets

An old open problem in graph drawing asks for the size of a universal point set, a set of points that can be used as vertices for straight-line drawings of all n-vertex planar graphs. We connect this problem to the theory of permutation patterns, where another open problem concerns the size of superpatterns, permutations that contain all patterns of a given size. We generalize superpatterns to ...

Bounds on Universal Sequences

Universal sequences for graphs, a concept introduced by Aleliunas [M. are studied. By letting U(d, n) denote the minimum length of a universal sequence for d-regular undirected graphs with n nodes, the latter paper has proved the upper bound U(d, n)= O(d2r log n) using a probabilistic argument. Here a lower bound of U(2, n)-(n log n) is proved from which U(d, n)-l'I(n log n) for all d is deduce...

Some Lower Bounds for Estrada Index

Upper and lower bounds for numerical radii of block shifts

For an n-by-n complex matrix A in a block form with the (possibly) nonzero blocks only on the diagonal above the main one, we consider two other matrices whose nonzero entries are along the diagonal above the main one and consist of the norms or minimum moduli of the diagonal blocks of A. In this paper, we obtain two inequalities relating the numeical radii of these matrices and also determine ...

Improved Lower Bounds for the Universal and a priori TSP

We consider two partial-information generalizations of the metric traveling salesman problem (TSP) in which the task is to produce a total ordering of a given metric space that performs well for a subset of the space that is not known in advance. In the universal TSP, the subset is chosen adversarially, and in the a priori TSP it is chosen probabilistically. Both the universal and a priori TSP ...