This paper describes an algorithm for coloring the nodes of a planar graph with no more than six colors using a self-stabilizing approach. The first part illustrates the coloring algorithm on a directed acyclic version of the given planar graph. The second part describes a self-stabilizing algorithm for generating the directed acyclic version of the planar graph, and combines the two algorithms into one. Keywords and Phrases: Self-stabilization, distributed algorithm, graph coloring, directed acyclic graph, atomicity. 1 Introduction Graph coloring is a classical problem in graph theory. The primary objective of graph coloring is to assign colors (from a given set) to the nodes of a graph, so that no two neighboring nodes have the same color. Of particular interest is the class of planar graphs, that has received substantial attention so far. A proof for 4-colorability already exists [1], and several algorithms [2] [5] [6] for node coloring are available with no more than five colors....

Abstract. For more than 2000 years, from Pythagoras and Euclid to Hilbert and Bourbaki, mathematical proofs were essentially based on axiomatic-deductive reasoning. In the last decades, the increasing length and complexity of many mathematical proofs led to the expansion of some empirical, experimental, psychological and social aspects, yesterday only marginal, but now changing radically the very essence of proof. In this paper, we try to organize this evolution, to distinguish its different steps and aspects, and to evaluate its advantages and shortcomings. Axiomatic-deductive proofs are not a posteriori work, a luxury we can marginalize nor are computer-assisted proofs bad mathematics. There is hope for integration! 1

We study vertex colorings of the square G 2 of an outerplanar graph G. We find the optimal bound of the inductiveness, chromatic number and the clique number of G 2 as a function of the maximum degree ∆ of G for all ∆ ∈ N. As a bonus, we obtain the optimal bound of the choosability (or the list-chromatic number) of G 2 when ∆ ≥ 7. In the case of chordal outerplanar graphs, we classify exactly which graphs have parameters exceeding the absolute minimum.

ization problems. Nonlinear programming did not receive as much attention in this respect until the recent work by Goemans and Williamson [62]. They use semidefinite programs, which are nonlinear programs, to obtain approximation solutions with much better approximation factors. For example, the best previously known approximation algorithm for MAXCUT, which was invented twenty years ago, has approximation factor 0.5 [137]. The algorithm of Goemans and Williamson dramatically improves the approximation factor to 0.878. Inspired by the work on MAXCUT, Karger, Motwani, and Sudan [86] adapt the same technique and obtain the currently best approximation algorithm for coloring a k-colorable graph with the fewest possible number of colors. The approximation ratio is improved by a factor of \Omega\Gamma n 2=k ) over the best previously known result [29]. Later Karger and Blum give the best known approximation algorithm for color

Whether ’tis nobler in the mind to suffer The slings and arrows of outrageous fortune, Or to take arms against a sea of troubles And by opposing end them? Hamlet 3/1, by W. Shakespeare In this paper we propose a new perspective on the evolution and history of the idea of mathematical proof. Proofs will be studied at three levels: syntactical, semantical and pragmatical. Computer-assisted proofs will be give a special attention. Finally, in a highly speculative part, we will anticipate the evolution of proofs under the assumption that the quantum computer will materialize. We will argue that there is little ‘intrinsic ’ difference between traditional and ‘unconventional ’ types of proofs. 2 Mathematical Proofs: An Evolution in Eight Stages Theory is to practice as rigour is to vigour. D. E. Knuth Reason and experiment are two ways to acquire knowledge. For a long time mathematical

The object of mathematical rigour is to sanction and