Coloring non-crossing strings

Louis Esperet 1 Daniel Gonçalves 2 Arnaud Labourel 3
1 G-SCOP_OC - OC
G-SCOP - Laboratoire des sciences pour la conception, l'optimisation et la production
2 ALGCO - Algorithmes, Graphes et Combinatoire
LIRMM - Laboratoire d'Informatique de Robotique et de Microélectronique de Montpellier
Abstract : For a family F of geometric objects in the plane, define χ(F) as the least integer such that the elements of F can be colored with colors, in such a way that any two intersecting objects have distinct colors. When F is a set of pseudo-disks that may only intersect on their boundaries, and such that any point of the plane is contained in at most k pseudo-disks, it can be proved that χ(F) 3k/2 + o(k) since the problem is equivalent to cyclic coloring of plane graphs. In this paper, we study the same problem when pseudo-disks are replaced by a family F of pseudo-segments (a.k.a. strings) that do not cross. In other words, any two strings of F are only allowed to " touch " each other. Such a family is said to be k-touching if no point of the plane is contained in more than k elements of F. We give bounds on χ(F) as a function of k, and in particular we show that k-touching segments can be colored with k + 5 colors. This partially answers a question of Hliněn´Hliněn´y (1998) on the chromatic number of contact systems of strings. * A preliminary version of this work appeared in the proceedings of EuroComb'09 [5].
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  • HAL Id : hal-01480244, version 1
  • ARXIV : 1511.03827

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Louis Esperet, Daniel Gonçalves, Arnaud Labourel. Coloring non-crossing strings. The Electronic Journal of Combinatorics, Open Journal Systems, 2016, 23 (4), pp.4.4. ⟨http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i4p4⟩. ⟨hal-01480244⟩

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