R. Bruce Richter

Cayley maps

We present a theory of Cayley maps, i.e., embeddings of Cayley graphs into oriented surfaces having the same cyclic rotation of generators around each vertex. These maps have often been used to encode symmetric embeddings of graphs. We also present an algebraic theory of Cayley maps and we apply the theory to determine exactly which regular or edge-transitive tilings of the sphere or plane are Cayley maps or Cayley graphs. Our main goal, however, is to provide the general theory so as to make it easier for others to study Cayley maps.

2-walks in circuit graphs

Abstract We prove the conjecture of Jackson and Wormald that every 3-connected planar graph has a closed walk visiting every vertex once or twice. This strengthens Barnette′s Theorem that every 3-connected planar graph has a spanning tree with maximum degree at most 3. The result also holds for 3-connected projective planar graphs.

Colouring Eulerian Triangulations

We show that for every orientable surface ? there is a number c so that every Eulerian triangulation of ? with representativeness ?c is 4-colourable.

The construction and classification of self-dual spherical polyhedra

Abstract In this paper we consider spherical polyhedra, or equivalently 3-connected embedded planar graphs. A self-duality map sends vertices to faces and faces to vertices while preserving incidence. We give six constructions of polyhedra with self-duality maps and show that these constructions yield all such polyhedra. Included is the construction of polyhedra which admit only self-duality maps of large order.

Separating and Nonseparating Disjoint Homotopic Cycles in Graph Embeddings

We show that if a graphGis embedded in a surface?with representativity?, thenGcontains at least ?(??1)/2? pairwise disjoint, pairwise homotopic, nonseparating (in?) cycles, andGcontains at least ?(??1)/8??1 pairwise disjoint, pairwise homotopic, separating, noncontractible cycles.

Triangular embeddings of complete graphs from graceful labellings of paths

We show that to each graceful labelling of a path on 2s+1 vertices, s>=2, there corresponds a current assignment on a 3-valent graph which generates at least 2^2^s cyclic oriented triangular embeddings of a complete graph on 12s+7 vertices. We also show that in this correspondence, two distinct graceful labellings never give isomorphic oriented embeddings. Since the number of graceful labellings of paths on 2s+1 vertices grows asymptotically at least as fast as (5/3)^2^s, this method gives at least 11^s distinct cyclic oriented triangular embedding of a complete graph of order 12s+7 for all sufficiently large s.

Hamiltonicity of 5-connected toroidal triangulations

In this article, we prove that a 5-connected toroidal triangulation has a contractible Hamilton cycle.

On the crossing numbers of certain generalized Petersen graphs

Abstract In his paper on the crossing numbers of generalized Peterson graphs, Fiorini proves that P(8,3) has crossing number 4 and claims at the end that P(10, 3) also has crossing number 4. In this article, we give a short proof of the first claim and show that the second claim is false. The techniques are interesting in that they focus on disjoint cycles, which must cross each other an even number of times.

The crossing number of K 3, n in a surface

In this article, we show that the crossing number of K3,n in a surface with Euler genus ϵ is ⌊n/(2ϵ + 2)⌋ (n − (ϵ + 1) {1 + ⌊n/(2ϵ + 2)⌋}). This generalizes a result of Guy and Jenkyns, who obtained this result for the torus.

The Crossing Number of P(N, 3)

Abstract. It is proved that the crossing number of the Generalized Petersen Graph P(3k+h,3) is k+h if h∈{0,2} and k+3 if h=1, for each k≥3, with the single exception of P(9,3), whose crossing number is 2.