Planar Graphs. Graphs on Higher Surfaces. Degrees. Critical Graphs. The Conjectures of Hadwiger and Hajos. Sparse Graphs. Perfect Graphs. Geometric and Combinatorial Graphs. Algorithms. Constructions. Edge Colorings. Orientations and Flows. Chromatic Polynomials. Hypergraphs. Infinite Chromatic Graphs. Miscellaneous Problems. Indexes.

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Graph Coloring Problems

This work deals with set-theoretic independence results (independence from the usual set-theoretic ZFC axioms), in particular for problems on the continuum. Consequently, the theory of iterated forcing is developed. The author gives a complete presentation of the theory of proper forcing and its relatives, starting from the beginning and avoiding the metamathematical considerations. In addition to particular consistency results, the author shows methods which can be used for such independence results. Many of these are presented in an "axiomatic" framework (a la Martin's axiom) for this reason. The main aim of this book is to enable a researcher interested in an independence result of the appropriate kind, to have much of the work done for him, thus allowing him to quote general results.

Proper and Improper Forcing

The term combinatorial Gray code was introduced in 1980 to refer to any method for generating combinatorial objects so that successive objects differ in some prespecified, small way. This notion generalizes the classical binary reflected Gray code scheme for listing n-bit binary numbers so that successive numbers differ in exactly one bit position, as well as work in the 1960s and 1970s on minimal change listings for other combinatorial families, including permutations and combinations.
The area of combinatorial Gray codes was popularized by Herbert Wilf in his invited address at the SIAM Conference on Discrete Mathematics in 1988 and his subsequent SIAM monograph [Combinatorial Algorithms: An Update, 1989] in which he posed some open problems and variations on the theme. This resulted in much recent activity in the area, and most of the problems posed by Wilf are now solved.
In this paper, we survey the area of combinatorial Gray codes, describe recent results, variations, and trends, and highlight some open problems.

A Survey of Combinatorial Gray Codes

General Theory: Hypergraphs. Fractional Matching. Fractional Coloring. Fractional Edge Coloring. Fractional Arboricity and Matroid Methods. Fractional Isomorphism. Fractional Odds and Ends. Appendix. Bibliography. Indexes.

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Fractional Graph Theory: A Rational Approach to the Theory of Graphs

We describe a fixed-point based approach to the theory of bipartite stable matchings. By this, we provide a common framework that links together seemingly distant results, like the stable marriage theorem of Gale and Shapley, the Mendelsohn-Dulmage theorem, the Kundu-Lawler theorem, Tarski's fixed-point theorem, the Cantor-Bernstein theorem, Pym's linking theorem, or the monochromatic path theorem of Sands et al. In this framework, we formulate a matroid-generalization of the stable marriage theorem and study the lattice structure of generalized stable matchings. Based on the theory of lattice polyhedra and blocking polyhedra, we extend results of Vande Vate and Rothblum on the bipartite stable matching polytope.

A fixed-point approach to stable matchings and some applications

On monochromatic paths in edge-coloured digraphs

Finite and Infinite Combinatorics in Sets and Logic

This paper concerns an intriguing conjecture involving vertex colorings of products of graphs. First we specify the product. Given (irreflexive, symmetric) graphs G and H, the product G x H has vertex set V(G) X V ( H ) and edges all pairs {(g, h), (g‘, h’)} such that gg‘ and hh’ are edges of G and H, respectively. We avoid modifying the label “product,” in part because of the variety of its names in the literature (e.g., Kronecker product, weak direct product, categorical product, conjunction, cardinal product). Given a coloring cp of G, there is a natural induced coloring p’ of G X H, namely cp’(g, h) = cp(g). Letting x, as usual, denote the chromatic number, we see that

On the chromatic number of the product of graphs

Two-colouring all two-element maximal antichains

For any positive integer k let B(k) denote the bipartite graph of k- and k+1-element subsets of a 2k+1-element set with adjacency given by containment. It has been conjectured that for all k, B(k) is Hamiltonian. Any Hamiltonian cycle would be the union of two (perfect) matchings. Here it is shown that for all k>1 no Hamiltonian cycle in B(k) is the union of two lexicographic matchings.

Bill Sands

Papers

On monochromatic paths in edge-coloured digraphs

Finite and Infinite Combinatorics in Sets and Logic

On the chromatic number of the product of graphs

Two-colouring all two-element maximal antichains

Lexicographic matchings cannot form Hamiltonian cycles