to appear in Combinatorics, Probability and Computing I show that the zeros of the chromatic polynomials PG(q) for the generalized theta graphs Θ (s,p) are, taken together, dense in the whole complex plane with the possible exception of the disc |q − 1 | < 1. The same holds for their dichromatic polynomials (alias Tutte polynomials, alias Potts-model partition functions) ZG(q,v) outside the disc |q + v | < |v|. An immediate corollary is that the chromatic roots of not-necessarily-planar graphs are dense in the whole complex plane. The main technical tool in the proof of these results is the Beraha–Kahane–Weiss theorem on the limit sets of zeros for certain sequences of analytic functions, for which I give a new and simpler proof. KEY WORDS: Graph, chromatic polynomial, dichromatic polynomial, Whitney rank function, Tutte polynomial, Potts model, Fortuin–Kasteleyn representation,

The so-called leapfrog transformation that was first introduced for fullerenes (trivalent polyhedra with 12 pentag-onal faces and all other faces hexagonal) is generalised to general polyhedra and maps on surfaces. All spher-ical polyhedra can be classified according to their leapfrog order. A polyhedron is said to be of Clar type if there exists a set of faces that cover each vertex exactly once. It is shown that a fullerene is of Clar type if and only if it is a leapfrog transform of another fullerene. Several basic transformations on maps are defined by means of which the leapfrog and other transformations can be accomplished. 1.

This is a historical survey of the Four Colour Theorem and a discussion of the philosophical implications of its proof. The problem, first stated as far back as 1850s, still causes controversy today. Its computeraided proof has forced mathematicians to question the notions of proofs and mathematical truth.

We present efficient algorithms for three coloring problems on subcubic graphs. (A subcubic graph has maximum degree at most three.) The first algorithm is for 4-edge coloring, or more generally, 4-list-edge coloring. Our algorithm runs in linear time, and appears to be simpler than previous ones. The second algorithm is the first randomized EREW PRAM algorithm for the same problem. It uses O(n= log n) processors and runs in O(log n) time with high probability, where n is the number of vertices of the graph.

We introduce and study the structure of an algebra in the linear space spanned by all regular 3-valent graphs with a prescribed order of edges at every vertex, modulo certain relations. The role of this object in various areas of low dimensional topology is discussed. 0 Introduction Regular graphs of degree 3, i. e. graphs in which every vertex is incident with exactly three edges, often occur in mathematics. Apart from graph theory proper, where such graphs are referred to as `cubic', they appear in a natural way in the topology of 3-manifolds, in the Vassiliev knot invariant theory and in connection with the four colour theorem. It turns out that in all these applications 3-valent graphs are endowed with a natural structure that consists in fixing, at every vertex of the graph, one of the two possible cyclic orders in the set of three edges issuing from this vertex. In the theory of Vassiliev invariants 3-valent graphs can be viewed as elements of the primitive subspace P of th...

We consider the minimum cut cover problem for a simple, undirected graphs G(V; E): find a minimum cardinality family of cuts C in G such that each edge e 2 E belongs to at least one cut C 2 C. The cardinality of the minimum cut cover of G is denoted by c(G). The motivation for this problem comes from testing of electronic component boards. Loulou has shown that the cardinality of a minimum cut cover in the complete graph is precisely dlog ne. However, determining the minimum cut cover of an arbitrary graph was posed as an open problem by Loulou. In this note we settle this open problem by showing that the cut cover problem is closely related to the graph coloring problem, thereby also obtaining a simple proof of Loulou's main result. We show that the problem is NP-complete in general, and moreover, the approximation version of this problem still remains NP-complete. Some other observations are made, all of which follow as a consequence of the close connection to graph coloring. Sup...

. We present a statement about Lie algebras that is equivalent to the Four Color Theorem. Contents 1. Introduction 1 1.1. Acknowledgement 3 2. Understanding W sl(N) 3 3. Understanding W sl(2) 5 References 7 1. Introduction Let us start by recalling a well-known construction that associates to any nite dimensional metrized Lie algebra L a numerical-valued functional W L dened on the set of all oriented trivalent graphs G (that is, trivalent graphs in which every vertex is endowed with a cyclic ordering of the edges emanating from it). This construction underlies the gaugegroup dependence of gauge theories in general and of the Chern-Simons topological eld theory in particular (see e.g. [B-N1, AS1, AS2]) and plays a prominent role in the theory of nite type (Vassiliev) invariants of knots ([B-N2, B-N3, B-N4]) and most likely also in the theory of nite type invariants of 3-manifolds ([O, GO, R]). Fix a nite dimensional metrized Lie algebra L (that is, a nite dimensional Lie algeb...

this paper we give a survey of the topics and results in topological graph theory. We offer neither breadth, as there are numerous areas left unexamined, nor depth, as no area is completely explored. Nevertheless, we do offer some of the favorite topics of the author and attempt to place them 1

The achromatic number problem is, given a graph G = (V, E), to find the greatest number of colors, Ψ(G), in a coloring of the vertices of G such that adjacent vertices get distinct colors and for every pair of colors some vertex of the first color and some vertex of the second color are adjacent. This problem is NP-complete even for trees. We obtain the following new results using combinatorial approaches to the problem. (1) A polynomial time O(|V | 3/8)-approximation algorithm for the problem on graphs with girth at least six. (2) A polynomial time 2-approximation algorithm for the problem on trees. This is an improvement over the best previous 7-approximation algorithm. (3) A linear time asymptotic 1.414-approximation algorithm for the problem when graph G is a tree with maximum degree d(|V |), where d: N − → N, such that d(|V |) = O(Ψ(G)). For example, d(|V |) = Θ(1) or d(|V |) = Θ(log |V |). (4) A linear time asymptotic 1.118-approximation algorithm for binary trees. We also improve the lower bound on the achromatic number of binary trees.

Topics: History, equivalent formulations and an outline of a proof. What are the prospects for finding a computer-free proof? Recommended reading: [1, 2, 3]