Let X Z/nZ denote the unitary Cayley graph of Z/nZ. We present results on the tightness of the known inequality γ(X Z/nZ ) ≤ γt(X Z/nZ ) ≤ g(n), where γ and γt denote the domination number and total domination number, respectively, and g is the arithmetic function known as Jacobsthal's function. In particular, we construct integers n with arbitrarily many distinct prime factors such that γ(X Z/nZ ) ≤ γt(X Z/nZ ) ≤ g(n) − 1. We give lower bounds for the domination numbers of direct products of complete graphs and present a conjecture for the exact values of the upper domination numbers of direct products of balanced, complete multipartite graphs.

No abstract

We consider the dynamics of light rays in triangle tilings where triangles are transparent and adjacent triangles have equal but opposite indices of refraction. We find that the behavior of a trajectory on a triangle tiling is described by an orientation-reversing threeinterval exchange transformation on the circle, and that the behavior of all the trajectories on a given triangle tiling is described by a polygon exchange transformation. We show that, for a particular choice of triangle tiling, certain trajectories approach the Rauzy fractal, under rescaling.

This paper has two aims: the first is to define a projective Fraissé family whose limit approximates the universal Knaster continuum. The family is such that the group Aut(K) of automorphisms of the Fraissé limit is a dense subgroup of the group, Homeo(K), of homeomorphisms of the universal Knaster continuum.The second aim is to compute the universal minimal flows of Aut(K) and Homeo(K). We prove that both have universal minimal flow homeomorphic to the universal minimal flow of the free abelian group on countably many generators. The computation involves proving that both groups contain an open, normal subgroup which is extremely amenable.

A star coloring of a graph G is a proper vertex coloring such that the subgraph induced by any pair of color classes is a star forest. The star chromatic number of G is the minimum number of colors needed to star color G. In this paper we determine the star-chromatic number of the splitting graphs of cycles of length n with n ≡ 1 (mod 3) and n = 5, resolving an open question of Furnmańczyk, Kowsalya, and Vernold Vivin.