Contract grant sponsor: Grant-In-Aid for Young Scientists (B) Contract grant number: 20740068; Contract grant sponsor: the Everett Pitcher Fund (to S. F.); Contract grant sponsor: NSF; Contract grant sponsor: ARC; Contract grant number: DP1096525 (to Á. S.).
Disconnected Colors in Generalized Gallai-Colorings
Version of Record online: 17 AUG 2012
© 2012 Wiley Periodicals, Inc.
Journal of Graph Theory
Volume 74, Issue 1, pages 104–114, September 2013
How to Cite
Fujita, S., Gyárfás, A., Magnant, C. and Seress, Á. (2013), Disconnected Colors in Generalized Gallai-Colorings. J. Graph Theory, 74: 104–114. doi: 10.1002/jgt.21694
- Issue online: 3 JUL 2013
- Version of Record online: 17 AUG 2012
- Manuscript Revised: 11 JUN 2012
- Manuscript Received: 16 OCT 2010
- Young Scientists (B). Grant Number: 20740068
- Everett Pitcher Fund
- ARC. Grant Number: DP1096525
Gallai-colorings of complete graphs—edge colorings such that no triangle is colored with three distinct colors—occur in various contexts such as the theory of partially ordered sets (in Gallai's original paper), information theory and the theory of perfect graphs. A basic property of Gallai-colorings with at least three colors is that at least one of the color classes must span a disconnected graph. We are interested here in whether this or a similar property remains true if we consider colorings that do not contain a rainbow copy of a fixed graph F. We show that such graphs F are very close to bipartite graphs, namely, they can be made bipartite by the removal of at most one edge. We also extend Gallai's property for two infinite families and show that it also holds when F is a path with at most six vertices.