%0 Journal Article
%T Strong oriented chromatic number of planar graphs without short cycles
%A Micka£¿l Montassier
%A Pascal Ochem
%A Alexandre Pinlou
%J Discrete Mathematics & Theoretical Computer Science
%D 2008
%I Discrete Mathematics & Theoretical Computer Science
%X Let M be an additive abelian group. A strong oriented coloringof an oriented graph G is a mapping ¦Õ from V(G) to M such that (1) ¦Õ(u) ¡Ù ¦Õ(v) whenever uv is an arc in G and (2) ¦Õ(v) - ¦Õ(u) ¡Ù -(¦Õ(t) - ¦Õ(z)) whenever uv and zt are two arcs in G. We say that G has a M-strong-oriented coloring. The strong oriented chromatic number of an oriented graph, denoted by ¦Ö s (G), is the minimal order of a group M, such that G has M-strong-oriented coloring. This notion was introduced by Ne et il and Raspaud. In this paper, we pose the following problem: Let i ¡Ý 4 be an integer. Let G be an oriented planar graph without cycles of lengths 4 to i. Which is the strong oriented chromatic number of G ? Our aim is to determine the impact of triangles on the strong oriented coloring. We give some hints of answers to this problem by proving that: (1) the strong oriented chromatic number of any oriented planar graph without cycles of lengths 4 to 12 is at most 7, and (2) the strong oriented chromatic number of any oriented planar graph without cycles of length 4 or 6 is at most 19.
%U http://www.dmtcs.org/dmtcs-ojs/index.php/dmtcs/article/view/455