 
Summary: Multicolored Forests in Bipartite Decompositions of Graphs
Noga Alon
IBM Almaden Research Center, San Jose, CA 95120
and
Sackler Faculty of Exact Sciences, Tel Aviv University, Israel
Richard A. Brualdi
Department of Mathematics, University of Wisconsin, Madison, WI 53706
Bryan L. Shader
Department of Mathematics, University of Wisconsin, Madison, WI 53706
Abstract
We show that in any edgecoloring of the complete graph Kn on n vertices, such that each
color class forms a complete bipartite graph, there is a spanning tree of Kn no two of whose
edges have the same color. This strengthens a theorem of Graham and Pollak and verifies a
conjecture of de Caen. More generally we show that in any edgecoloring of a graph G with p
positive and q negative eigenvalues, such that each color class forms a complete bipartite graph,
there is a forest of at least max{p, q} edges no two of which have the same color. In case G is
bipartite there is always such a forest which is a matching.
Research partially supported by National Science Foundation Grant No. DMS8901445 and National Security
Agency Grant No. MDA90489H2060
