The exceptional Dehn filling conjecture of the second author concerning the relationship between exceptional slopes α, β on the boundary of a hyperbolic knot manifold M has been verified in all cases other than small Seifert filling slopes. In this paper we verify it when α is a small Seifert filling slope and β is a toroidal filling slope in the generic case where M admits no punctured-torus f...

The chromatic polynomials of some families of quadrangulations of the torus can be found explicitly. The method, known as ‘bracelet theory’ is based on a decomposition in terms of representations of the symmetric group. The results are particularly appropriate for studying the limit curves of the chromatic roots of these families. In this paper these techniques are applied to a family of quadra...

When the standard representation of a crystallographic Coxeter group Γ is reduced modulo an odd prime p, one obtains a finite group G acting on some orthogonal space over Zp. If Γ has a string diagram, then G p will often be the automorphism group of a finite abstract regular polytope. In parts I and II we established the basics of this construction and enumerated the polytopes associated to gr...

Suppose G is a graph embedded in S g with width (also known as edge width) at least 264(2 g ?1). If P V (G) is such that the distance between any two vertices in P is at least 16, then any 5-coloring of P extends to a 5-coloring of all of G. We present similar extension theorems for 6-and 7-chromatic toroidal graphs, for 3-colorable large-width graphs embedded on S g with every face even sided,...

A set S V is a dominating set of a graph G = (V;E) if each vertex in V is either in S or is adjacent to a vertex in S. A vertex is said to dominate itself and all its neighbors. The domination number (G) is the minimum cardinality of a dominating set of G. A set S V is an independent set of vertices if no two vertices in S are adjacent. The independence number, B0 (G), is the maximum cardinalit...

Abstract. The average effective resistance of a graph is a relevant performance index in many applications, including distributed estimation and control of network systems. In this paper, we study how the average resistance depends on the graph topology and specifically on the dimension of the graph. We concentrate on d-dimensional toroidal grids and we exploit the connection between resistance...

A radio coloring of a simple connected graph G is a mapping f : V (G) → {0, 1, 2, . . .} such that |f(u)− f(v)| diam(G)+1−d(u, v) for each pair of distinct vertices u and v of G, where diam(G) is the diameter of G and d(u, v) is the distance between u and v in G. The span of a radio coloring f , span(f), is the number maxu∈V (G) f(u). The radio number of G, rn(G), is defined as minf{span(f) : f...

In this paper, we focus on the design of network topology to achieve fast information distribution. We present the information distribution performance of Borel Cayley graphs, a family of pseudo-random graphs, is far superior than that of other well-known graph families. To demonstrate the effectiveness of this pseudo-random approach, we compare the convergence speed of the average consensus pr...

We describe how some simple properties of discrete one-forms directly relate to some old and new results concerning the parameterization of 3D mesh data. Our first result is an easy proof of Tutte’s celebrated “springembedding” theorem for planar graphs, which is widely used for parameterizing meshes with the topology of a disk as a planar embedding with a convex boundary. Our second result gen...

In this paper we consider the spanning tree congestion for several families of graphs. We find the exact spanning tree congestion for toroidal meshes, Cm×Cn, and cylindrical meshes, Pm×Cn,. Also we find bounds for the spanning tree congestion of Qn, and a construction that gives the upper bound.