Rainbow Numbers for Cycles in Plane Triangulations Mirko Hor ˇ n´ ak, 1 Stanislav Jendrol’, 1 Ingo Schiermeyer, 2 and Roman Sot ´ ak 1 1 INSTITUTE OF MATHEMATICS P. J. ˇ SAF ´ ARIK UNIVERSITY KO ˇ SICE SLOVAKIA E-mail: mirko.hornak@upjs.sk; stanislav.jendrol@upjs.sk; roman.sotak@upjs.sk 2 INSTITUT F ¨ UR DISKRETE MATHEMATIK UND ALGEBRA TECHNISCHE UNIVERSIT ¨ AT BERGAKADEMIE FREIBERG FREIBERG, GERMANY E-mail: ingo.schiermeyer@tu-freiberg.de Received May 22, 2013; Revised March 26, 2014 Published online in Wiley Online Library (wileyonlinelibrary.com). DOI 10.1002/jgt.21803 Abstract: In the article, the existence of rainbow cycles in edge col- ored plane triangulations is studied. It is shown that the minimum number rb(T n , C 3 ) of colors that force the existence of a rainbow C 3 in any n-vertex plane triangulation is equal to ⌊ 3n−4 2 ⌋. For k ≥ 4 a lower bound and for k ∈{4, 5} an upper bound of the number rb(T n , C k ) is determined. C  2014 Wiley Periodicals, Inc. J. Graph Theory 00: 1–10, 2014 Keywords: edge coloring; rainbow number; rainbow subgraph; triangulation Contract grant sponsor: Slovak Science and Technology Assistance Agency; Con- tract grant number: APVV-0023-10; Contract grant sponsor: Slovak VEGA; Contract grant number: 1/0652/12; Contract grant sponsor: P. J. ˇ Saf ´ arik University within the project EXPERT; Contract grant number: ITMS 26110230056. Journal of Graph Theory C  2014 Wiley Periodicals, Inc. 1