Dan Archdeacon

A kuratowski theorem for the projective plane

An embedding of a graph G into a surface S is a realization of G as a subspace of S . A graph G is irreducible for S if G does not embed in S , but any proper subgraph of G does embed in S. Irreducible graphs are the smallest (with respect to containment) graphs which fail to embed on a given surface. Let I ( S ) denote the set of graphs, each with no valency 2 vertices, which are irreducible for S . Using this notation we state Kuratowski’s theorem [ 71:

Maximum genus and connectivity

Abstract It is shown that ⌈β(G)/3⌉ is the tight lower bound on the maximum genus γM(G) of 2-edge-connected simplicial graphs, where β(G) is the cycle rank of the graph G. Also, a systematic method is developed to construct 3-vertex-connected simplicial graphs G satisfying the equality γM(G) = ⌈β(G)/3⌉. These two results combine with previously known results to yield a complete picture of the tight lower bounds on the maximum genus of simplicial graphs.

Some remarks on domination: SOME REMARKS ON DOMINATION

Proof: The proof is by induction on |V (G)| + |E(G)|. The smallest graph as described in the lemma is K1,3, for which the statement holds. This gives the start of our induction. Let x = |X| and y = |Y |. If there exists a vertex v in Y of degree at least 4, then delete any edge e incident to v. The subset A of G− e guaranteed by the inductive hypothesis is adjacent in G to every vertex in Y as desired. So we may assume that the vertices in Y are all of

A Kuratowski theorem for nonorientable surfaces

Abstract Let Σ denote a surface. A graph G is irreducible for Σ provided that G does not embed in Σ, but any proper subgraph does so embed. Let I ( Σ ) denote the set of graphs without degree two vertices which are irreducible for Σ. Observe that a graph embeds in Σ if and only if it does not contain a subgraph homeomorphic to a member of I ( Σ ). For example, Kuratowskis theorem shows that I ( Σ ) = { K 3,3 , K 5 } when Σ is the sphere. In this paper we prove that the set I ( Σ ) is finite for each nonorientable surface, setting in part a conjecture of Erdos from the 1930s.

A note on defective colorings of graphs in surfaces

A graph is (m, k)-colorable if its vertices can be colored with m colors in such a way that each vertex is adjacent to at most k vertices of the same color as itself. In a recent paper Cowen, Cowen, and Woodall proved that, for each compact surface S, there exists an integer k = k(S) such that every graph in S can be (4, k)-colored. They also conjectured that the 4 could be replaced by 3. In this note we prove their conjecture.

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.

Fractional chromatic numbers of cones over graphs

It has been shown that every quadrangulation on any nonspherical orientable closed surface with a suf®ciently large representativity has chromatic number at most 3. In this paper, we show that a quadrangulation G on a nonorientable closed surface Nk has chromatic number at least 4 if G has a cycle of odd length which cuts open Nk into an orientable surface. Moreover, we characterize the quadrangulations on the torus and the Klein bottle with chromatic number exactly 3. By our characterization, we prove that every quadrangulation on the torus with representativity at least 9 has chromatic number at most 3, and that a quadrangulation on the Klein bottle with representativity at least 7 has chromatic number at most 3 if a cycle cutting open the Klein bottle into an annulus has even length. As an application of our theory, we prove that every nonorientable closed surface Nk admits an eulerian triangulation with chromatic number at least 5 which has arbitrarily large representativity. ß 2001 John Wiley & Sons, Inc. J Graph Theory 37: 100±114, 2001

The medial graph and voltage-current duality

Abstract The theories of current graphs and voltage graphs give powerful methods for constructing graph embeddings and branched coverings of surfaces. Gross and Alpert first showed that these two theories were dual, that is, that a current assignment on an embedded graph was equivalent to a voltage assignment on the embedded dual. In this paper we examine current and voltage graphs in the context of the medial graph, a 4-regular graph formed from an embedded graph which encodes both the primal and dual graphs. As a consequence we obtain new insights into voltage-current duality, including wrapped coverings. We also develop a method for simultaneously giving a voltage and a current assignment on an embedded graph in the case that the voltage-current group is abelian. We apply this technique to construct self-dual embeddings for a variety of graphs. We also construct orientable and non-orientable embeddings of K p , q with dual K r , s for all possible p , q , r , s even with pq = rs .

How to Exhibit Toroidal Maps in Space

AbstractSteinitzs theorem states that a graph is the 1-skeleton of a convex polyhedron if and only if it is 3-connected and planar. The polyhedron is called a geometric realization of the embedded graph. Its faces are bounded by convex polygons whose points are coplanar. A map on the torus does not necessarily have such a geometric realization. In this paper we relax the condition that faces are the convex hull of coplanar points. We require instead that the convex hull of the points on a face can be projected onto a plane so that the boundary of the convex hull of the projected points is the image of the boundary of the face. We also require that the interiors of the convex hulls of different faces do not intersect. Call this an exhibition of the map. A map is polyhedral if the intersection of any two closed faces is simply connected. Our main result is that every polyhedral toroidal map can be exhibited. As a corollary, every toroidal triangulation has a geometric realization.

Densely embedded graphs

Abstract Let G be a graph embedded in a surface S . The face-width of the embedding is the minimum size | C ⋔ G | over all noncontractible cycles C in S . The face-width measures how densely a graph is embedded in a surface, equivalently, how well an embedded graph represents a surface. In this paper we present a construction of densely embedded graphs with a variety of interesting properties. The first application is the construction of embeddings where both the primal and the dual graph have large girth. A second application is the construction of a graph with embeddings on two different surfaces, each embedding of large face-width. These embeddings are counterexamples to a conjecture by Robertson and Vitray. In the third application we examine the number of triangles needed to triangulate a surface S such that every noncontractible cycle is of length at least k . Surprisingly, for large g the number is approximately 4 g , regardless of k . The fourth application is the construction of trivalent polygonal graphs. We close with some observations and directions for further research.