Rainbow spanning trees in complete graphs colored by one‐factorizations

Abstract

Brualdi and Hollingsworth conjectured that, for even n, in a proper edge coloring of Kn using precisely n−1 colors, the edge set can be partitioned into n/2 spanning trees which are rainbow (and hence, precisely one edge from each color class is in each spanning tree). They proved that there always are two edge disjoint rainbow spanning trees. Krussel, Marshall, and Verrall improved this to three edge disjoint rainbow spanning trees. Recently, Carraher, Hartke and the author proved a theorem improving this to εnlogn rainbow spanning trees, even when more general edge colorings of Kn are considered. In this article, we show that if Kn is properly edge colored with n−1 colors, a positive fraction of the edges can be covered by edge disjoint rainbow spanning trees.

Decompositions into spanning rainbow structures

“…These conjectures attracted a lot of attention from various researchers (see, for example, ) who showed how to find several disjoint spanning rainbow trees. The best known results for these problem guarantee the existence of edge‐disjoint rainbow trees (see for Conjecture and for Conjecture ). Developing our results on Hamiltonian cycles, we are able to improve this and show that one can find disjoint spanning rainbow trees.…”

On the number of rainbow spanning trees in edge-colored complete graphs

“…In this paper, we focus on Conjecture 1 by proving that in any (2m − 1)-edge-coloring of K 2m , m ≥ 1, there exist at least √ 6m+9 3 mutually edge-disjoint rainbow spanning trees. Asymptotically, this is not as good as the bound in [6] or [10], but our result applies to all values of m and it is better until m is extremely large (over 5.7 × 10 7 for the bound in [6]). Instead of using the probabilistic method to prove the result, as was used in [6] and [10], we derive our bound by means of an explicit, constructive approach.…”

“…Asymptotically, this is not as good as the bound in [6] or [10], but our result applies to all values of m and it is better until m is extremely large (over 5.7 × 10 7 for the bound in [6]). Instead of using the probabilistic method to prove the result, as was used in [6] and [10], we derive our bound by means of an explicit, constructive approach. So, not only do we actually produce the rainbow trees, but also some structure of each rainbow spanning tree is determined in the process.…”

“…First, from f (Ω) and the definition of L Ω−1 , we can follow the same steps as we did in (7) to see that |L Ω−1 | ≥ 2m − 3Ω 2 + 6Ω − 1 and further, that L * Ω−1 ≥ 2m − 3Ω 2 + 6Ω − 3. Now, since by the induction hypothesis k ≤ Ω and by (10) and (14), L * i−1 > |L * i | for 2 ≤ i ≤ k − 1, we have the following by our choice of Ω:…”

“…Kaneko, Kano, and Suzuki [12] then improved the previous result slightly by showing that three edge-disjoint rainbow spanning trees exist in any proper edge-coloring of K 2m . Recently, Horn [10] showed that for m sufficiently large there is an ǫ > 0 such that every properly (2m − 1)-edge-colored K 2m has ǫ2m edge-disjoint rainbow spanning trees. And a recently submitted paper by Pokrovskiy and Sudakov [14] shows that every properly (n − 1)-edge-colored K n has n 9 edge-disjoint spanning rainbow trees.…”