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.

/pdf/graph-coloring-problems-4v7vijos3s.pdf

Graph Coloring Problems

The energy of a graph

Kneser's conjecture, chromatic number, and homotopy

I was asked to write a paper about the major unsolved problems in com-binatorial mathematics. After some thought it seemed better to modify the title to a less pretentious one. Combinatorial mathematics has grown enormously and a genuine survey would have to include not only topics where I have no real competence but also topics about which I n e v er seriously thought, e.g. algorithmic combinatorics, coding theory and matroid theory. There is no doubt that the proof of the conjecture that several simply stated problems have no good algorithm is fundamental and may h a ve i m-portant consequences for many other branches of mathematics, but unfortunately I have no real feeling for these questions and I feel I should leave the subject to those who are more competent. I just heard that Khachiyan 59], has a polynomial algorithm for linear programming. (See also 50].) This is considered a sensational result and during my last stay i n the U.S. many o f m y friends were greatly impressed by i t. First of all I will discuss some problems on set systems. I state only my three favourite problems, but before starting I refer to the survey paper 31].

On the combinatorial problems which I would most like to see solved

http://homepages.math.uic.edu/~suk/erdos_hajnal.pdf

Ramsey-type theorems

On chromatic number of graphs and set-systems

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On chromatic number of graphs and set-systems