Zero Sums on Unit Square Vertex Sets and Plane Colorings

On the Defining Number of (2n - 2)-Vertex Colorings of Kn x Kn

Innnite Paths with Bounded or Recurrent Partial Sums

We consider problems of the following type. Assign independently to each vertex of the square lattice the value +1, with probability p, or ?1, with probability 1 ? p. We ask whether an innnite path exists, with the property that the partial sums of the 1s along are uniformly bounded, and whether there exists an innnite path with the property that the partial sums along are equal to zero innnite...

Strong parity vertex coloring of plane graphs

Czap and Jendrol’ introduced the notions of strong parity vertex coloring and the corresponding strong parity chromatic number χs. They conjectured that there is a constant bound K on χs for the class of 2-connected plane graphs. We prove that the conjecture is true with K = 97, even with an added restriction to proper colorings. Next, we provide simple examples showing that the sharp bound is ...

Extremal H-colorings of trees and 2-connected graphs

For graphs G and H, an H-coloring of G is an adjacency preserving map from the vertices of G to the vertices of H. H-colorings generalize such notions as independent sets and proper colorings in graphs. There has been much recent research on the extremal question of finding the graph(s) among a fixed family that maximize or minimize the number of H-colorings. In this paper, we prove several res...

Cyclic 4-Colorings of Graphs on Surfaces

To attack the Four Color Problem, in 1880, Tait gave a necessary and sufficient condition for plane triangulations to have a proper 4-vertex-coloring: a plane triangulation G has a proper 4-vertex-coloring if and only if the dual of G has a proper 3-edge-coloring. A cyclic coloring of a map G on a surface F 2 is a vertex-coloring of G such that any two vertices x and y receive different colors ...