Let G be a group. The permutability graph of subgroups of G, denoted by Γ(G), is a graph having all the proper subgroups of G as its vertices, and two subgroups are adjacent in Γ(G) if and only if they permute. In this paper, we classify the finite groups whose permutability graphs are toroidal or projective-planar. In addition, we classify the finite groups whose permutability graph does not c...

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.

A graph is (k; d)-colorable if one can color the vertices with k colors such that no vertex is adjacent to more than d vertices of the same color. In this paper we investigate the existence of such colorings in surfaces. It is shown that a toroidal graph is (3; 2)and (5; 1)colorable, and that a graph of genus is ( =(d + 1) + 4; d)-colorable, where is the maximum chromatic number of a graph embe...

Recently, the author found that there is a common mistake in some papers by using minimal counterexample and discharging method. We first discuss how the mistake is generated, and give a method to fix the mistake. As an illustration, we consider total coloring of planar or toroidal graphs, and show that: if G is a planar or toroidal graph with maximum degree at most κ − 1, where κ ≥ 11, then th...

Forbidden minors and subdivisions for toroidal graphs are numerous. In contrast, the toroidal graphs with no K3,3’s have a nice explicit structure and short lists of obstructions. For these graphs, we provide the complete lists of four forbidden minors and eleven forbidden subdivisions.

We consider the problem of nding a minimum cost acceptable path on a toroidal grid graph, where each horizontal and each vertical edge have the same orientation. An acceptable path is closed path that makes a complete horizontal and vertical circuit. We exploit the structure of this graph to develop eecient parallel algorithms for a message passing computer. Given p processors and an m by n tor...

In this paper, we build on the work of Alspach, Chen, and Dean [2] who showed that proving the hamiltonicity of the Cayley graph of the the dihedral group Dn reduces to showing that certain cubic, connected, bipartite graphs (called honeycomb toroidal graphs) are hamilton laceable; that is, any two vertices at odd distance from each other can be joined by a hamilton path. Alspach, Chen, and Dea...