Nora Hartsfield

Nonhamiltonian Triangulations with Large Connectivity and Representativity

In this paper we show that for eachnthere is a triangulation of an orientable surface which isn-connected,n-representative, and such that every spanning tree contains a vertex of degree at leastn. This also shows that there exist such graphs where any spanning walk must visit some vertex at leastntimes. We show the analogous results for nonorientable surfaces.

A family of sparse graphs of large sum number

Given an integer r > 0, let Gr, = (Vr, E) denote a graph consisting of a simple finite undirected graph G = (V, E) of order n and size m together with r isolated vertices . Then | V | = n, |Vr| = n+r, and |E| = m. Let L:Vr → + denote a labelling of the vertices of Gr with distinct positive integers. Then Gr is said to be a sum graph if there exists a labelling L such that for every distinct vertex pair u and v of Vr, (u, v) ϵE if and only if there exists a vertex wϵVr whose label L(w) = L(u) + L(v). For a given graph G, the sum numberσ = σ(G) is defined to be the least value of r for which Gr is a sum graph. Gould and Rodl have shown that there exist infinite classes of graphs such that, over Gϵ, σ(G)ϵΘ(n2), but no such classes have been constructed. In fact, for all classes for which constructions have so far been found, σ(G)ϵo(m). In this paper we describe constructions which show that for wheels Wn of (sufficiently large) order n + 1 and size m = 2n, σ(Wn) = n/2 + 3 if n is even and n ⩽ σ (Wn) ⩽ n + 2 if n is odd. Hence for wheels σ (Wn) ϵΘ(m).

The splitting number of the complete graph

If a given graphG can be obtained bys vertex identifications from a suitable planar graph ands is the minimum number for which this is possible thens is called the splitting number ofG. Here a formula for the splitting number of the complete graph is derived.

Obstructions to directed embeddings of Eulerian digraphs in the plane

A 2-cell embedding of an Eulerian digraph in a closed surface is said to be directed if the boundary of each face is a directed closed walk in G. We prove Kuratowski-type theorems about obstructions to directed embeddings of Eulerian digraphs in the plane.

The toroidal splitting number of the complete graph K n

Abstract The splitting number of a graph on a surface is defined, and a formula for the splitting number of the complete graph on the torus is established. The method of proof is to use Eulers generalized polyhedral formula for the lower bound, then current graphs are used to obtain the duals of splittings of K 12 s +1 and K 12 s +7 . Suitable additions are made to these duals to obtain the other cases.

Self-dual embeddings of complete bipartite graphs

Abstract In this paper we examine self-dual embeddings of the complete bipartite graph K n , m on both orientable and nonorientable surfaces. It can be quickly established that such an embedding can exist only if both n and m are even integers exceeding 2. We show by construction that these necessary conditions are sufficient, except that there is no orientable self-dual embedding of K 6,6 .