By a drawing of a graph G, we mean a drawing in the plane such that vertices are represented by distinct points and edges by arcs. The crossing number cr(G) of a graph G is the minimum possible number of crossings in a drawing of G. The pair-crossing number pair-cr(G) of G is the minimum possible number of (unordered) crossing pairs in a drawing of G. Clearly, pair-cr(G) ≤ cr(G) holds for any g...

We examine several types of visibility graphs: bar and semi-bar k-visibility graphs, rectangle k-visibility graphs, arc and circle k-visibility graphs, and compact visibility graphs. We improve the upper bound on the thickness of bar k-visibility graphs from 2k(9k − 1) to 6k, and prove that the upper bound must be at least k + 1. We also show that the upper bound on the thickness of semi-bar k-...

Let K be an algebraic number field. Assume that ζK(s)/ζ(s) is entire. We give an explicit upper bound for the residue at s = 1 of the Dedekind zeta function ζK(s) of K. We deduce explicit upper bounds on class numbers of cubic and quartic number fields.

A number of upper and lower bounds are obtained for K( n, R), the minimal number of codewords in any binary code of length n and covering radius R. Several new constructions are used to derive the upper bounds, including an amalgamated direct sum construction for nonlinear codes. This construction works best when applied to normal codes, and we give some new and stronger conditions which imply ...

Given a graph G and a positive integer k, define the Gallai-Ramsey number to be the minimum number of vertices n such that any k-edge-coloring of Kn contains either a rainbow (all different colored) triangle or a monochromatic copy of G. In this work, we improve upon known upper bounds on the Gallai-Ramsey numbers for paths and cycles. All these upper bounds now have the best possible order of ...

The upper chromatic number (H) of a hypergraphH= (X,E) is the maximum number k for which there exists a partition of X into non-empty subsets X=X1 ∪X2 ∪ · · · ∪Xk such that for each edge at least two vertices lie in one of the partite sets. We prove that for every n 3 there exists a 3-uniform hypergraph with n vertices, upper chromatic number 2 and n(n− 2)/3 edges which implies that a correspon...

This paper provides lower orientable k-step domination number and upper orientable k-step domination number of complete r-partite graph for 1 ≤ k ≤ 2. It also proves that the intermediate value theorem holds for the complete r-partite graphs.