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,

In this paper I try to present my field, combinatorics, via five examples of combinatorial studies which have some geometric flavor. The first topic is Tverberg's theorem, a gem in combinatorial geometry, and various of its combinatorial and topological extensions. McMullen's upper bound theorem for the face numbers of convex polytopes and its many extensions is the second topic. Next are general properties of subsets of the vertices of the discrete n-dimensional cube and some relations with questions of extremal and probabilistic combinatorics. Our fourth topic is tree enumeration and random spanning trees, and finally, some combinatorial and geometrical aspects of the simplex method for linear programming are considered.

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.

A graph is a minor of another if the first can be obtained from a subgraph of the second by contracting edges. An excluded minor theorem describes the structure of graphs with no minor isomorphic to a prescribed set of graphs. Splitter theorems are tools for proving excluded minor theorems. We discuss splitter theorems for internally 4-connected graphs and for cyclically 5-connected cubic graphs, the graph minor theorem of Robertson and Seymour, linkless embeddings of graphs in 3-space, Hadwiger’s conjecture on t-colorability of graphs with no Kt+1 minor, Tutte’s edge 3-coloring conjecture on edge 3-colorability of 2-connected cubic graphs with no Petersen minor, and Pfaffian orientations of bipartite graphs. The latter are related to the even directed circuit problem, a problem of Pólya about permanents, the 2-colorability of hypergraphs, and sign-nonsingular matrices.

In this paper we have shown without assuming the four color theorem of planar graphs that every (bridgeless) cubic planar graph has a threeedge-coloring. This is an old-conjecture due to Tait in the squeal of efforts in settling the four-color conjecture at the end of the 19th century. We have also shown the applicability of our method to another well-known three edge-coloring conjecture on cubic graphs. Namely Tutte’s conjecture that ”every 2-connected cubic graph with no Petersen minor is 3-edge colorable”. Hence the conclusion of this paper implies another non-computer proof of the four color theorem by using spiral-chains in different context. 1

We discuss splitter theorems for internally 4-connected graphs and for cyclically 5-connected cubic graphs, the graph minor theorem, linkless embeddings, Hadwiger's conjecture, Tutte's edge 3-coloring conjecture, and Pfaffian orientations of bipartite graphs.

. Coloring problems include some of the most classical combinatorial problems (such as map coloring problems or edge coloring of cubic graphs). But the area is alive and well beeing refreshed by a constant stream of new problems and solutions of old ones. In this paper we complement this by a survey of four particular problems which are all related to the notion of a homomorphism. This forms one of the most recent approaches to coloring problems. Keywords. Chromatic number, Planar graphs, Homomorphism, Gap, Density, Quasiordering, Oriented chromatic number, Game chromatic number. 1

The Temperley-Lieb algebra Tn with parameter 2 is the associative algebra over Q generated by 1, e0, el,..., en, where the generators satisfy the rela- 2 = 2el, eiejei = ei if [i- j[ = 1 and eiej = ejei if [i- j[ > 2. We tions i - use the Four Color Theorem to give a necessary and sufficient condition for certain elements of Tn to be nonzero. It turns out that the characterization is, in fact, equivalent to the Four Color Theorem.

This compendium is the result of reformatting and minor editing of the author's transparencies for his talk delivered at the conference. The talk covered Euler's Formula, Kuratowski's Theorem, linear planarity tests, Schnyder's Theorem and drawing on the grid, the two paths problem, Pfaffian orientations, linkless embeddings, and the Four Color Theorem.

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]