We give an exact characterization of 3-colorability triangle-free graphs drawn in the torus, form 186 “templates” (graphs with certain faces filled by arbitrary quadrangulations) such that a graph from this class is not 3-colorable if and only it contains subgraph matching one templates. As consequence, we show every torus edge-width at least six 3-colorable, key property used efficient algorit...

A toroidal periodic graph G D is defined by an integral d = d matrix D and a directed graph G in which the edges are associated with d-dimensional integral vectors. The periodic graph has a vertex for each vertex of the static graph and for each integral position in the parallelpiped defined by the columns of D. There is an edge from vertex u at position y to vertex ̈ at position z in the period...

We prove a precise characterization of toroidal graphs with no K3,3-subdivisions in terms of forbidden minors and subdivisions. The corresponding lists of four forbidden minors and eleven forbidden subdivisions are shown.

We consider the degree/diameter problem for graphs embedded in a surface, namely, given a surface Σ and integers ∆ and k, determine the maximum order N(∆, k,Σ) of a graph embeddable in Σ with maximum degree ∆ and diameter k. We introduce a number of constructions which produce many new largest known planar and toroidal graphs. We record all these graphs in the available tables of largest known ...

Abstmct. Wagner’s theorem (any two maximal plane graphs having p vertices are equivalent under diagonal transformations) is extended to maximal torus graphs, graphs embedded in the torus with a maximal set of edges present. Thus any maximal torus graph havingp vertices may be diagonally transformed into any other maximal torus p;aph having p vertices. As with Wag mr’s theorem, a normal form rep...

It is well known that not all 3-connected planar graphs are hamiltonian. Whitney [10] proved that every triangulation of the sphere with no separating triangles is hamiltonian. Tutte [9] proved that every 4-connected planar graph has a Hamilton cycle. Extending Tutte's technique, Thomassen [8] proved that every 4-connected planar graph is in fact Hamilton connected. (A small omission in [8] was...

In the final project paper we consider a graph parameter called readability. Motivation for readability comes from bioinformatics applications. Graphs arising in problems related to genome sequencing are of small readability, which motivates the study of graphs of small readability. We present an algorithm due to Braga and Meidanis, which shows that every digraph is isomorphic to the overlap gr...

For each positive integer n, the toroidal queens graph may be described as a graph with vertex set Zn × Zn where every vertex is adjacent to those vertices in the directions (1, 0), (0, 1), (1, 1), (1,−1) from it. We here extend this idea, examining graphs with vertex set F × F , where F is a finite field, and any four directions are used to define adjacency. The automorphism groups and isomorp...

In 1974, Kundu [4] has shown that triangulated (or maximal) simple toroidal graphs have three edge-disjoint spanning trees. We extend this result by showing that a triangulated graph on an orientable surface of genus g has at least three edge-disjoint spanning trees and so we can partition the edges of graphs of genus g into three forests plus a set of at most 6g − 3 edges.