Odd Wheels Are Not Odd-Distance Graphs

An odd wheel graph is a graph formed by connecting a new vertex to all vertices of an odd cycle. We answer a question of Rosenfeld and Le by showing that odd wheels cannot be drawn in the plane so that the lengths of the edges are odd integers.

Chromatic Coloring of Distance Graphs I

The primary aim of this paper is to publicize various problems regarding chromatic coloring of finite, simple and undirected graphs. A simple motivation for this work is that the coloring of graphs gives models for a variety of real world problems such as scheduling. We prove some interesting results related to the computation of chromatic number of certain distance graphs and also discuss some open problems.

Density of sets with missing differences and applications

For a given set M of positive integers, a well-known problem of Motzkin asks to determine the maximal asymptotic density of M-sets, denoted by μ(M), where an M-set is a set of non-negative integers in which no two elements differ by an element in M. In 1973, Cantor and Gordon find μ(M) for |M| ≤ 2. Partial results are known in the case |M| ≥ 3 including some results in the case when M is an infinite set. Motivated by some 3 and 4-element families already discussed by Liu and Zhu in 2004, we study μ(M) for two families namely, M = {a, b,a + b, n(a + b)} and M = {a, b, b − a, n(b − a)}. For both of these families, we find some exact values and some bounds on μ(M). This number theory problem is also related to various types of coloring problems of the distance graphs generated by M. So, as an application, we also study these coloring parameters associated with these families.