Multicolored trees in complete graphs

Abstract

AbstractA multicolored tree is a tree whose edges have different colors. Brualdi and Hollingsworth 5 proved in any proper edge coloring of the complete graph K2n(n > 2) with 2n − 1 colors, there are two edge‐disjoint multicolored spanning trees. In this paper we generalize this result showing that if (a1,…, ak) is a color distribution for the complete graph Kn, n ≥ 5, such that $2 \leq a_{1} \leq a_{2} \leq \cdots \leq a_{k} \leq (n+ 1)/ 2$, then there exist two edge‐disjoint multicolored spanning trees. Moreover, we prove that for any edge coloring of the complete graph Kn with the above distribution if T is a non‐star multicolored spanning tree of Kn, then there exists a multicolored spanning tree T' of Kn such that T and T' are edge‐disjoint. Also it is shown that if Kn, n ≥ 6, is edge colored with k colors and $k \geq {{{n}-{2}} \choose {2}}+{3}$, then there exist two edge‐disjoint multicolored spanning trees. © 2006 Wiley Periodicals, Inc. J Graph Theory 54: 221–232, 2007