Partitioning 3-edge-colored complete equi-bipartite graphs by monochromatic trees under a color degree condition

TL;DR

This paper proves that in a 3-edge-colored complete bipartite graph with each vertex having all three colors, the vertices can be covered by at most three monochromatic trees, extending understanding of monochromatic tree partitions.

Contribution

It establishes that for a 3-edge-colored complete bipartite graph with each vertex having all three colors, the monochromatic tree partition number is exactly three.

Findings

For n ≥ 3, in a 3-edge-colored K(n,n) with each vertex having color degree 3, t_3(K(n,n))=3.

The result extends the understanding of monochromatic tree partitions in bipartite graphs.

The paper provides an exact value for the monochromatic tree partition number under specific coloring conditions.

Abstract

The monochromatic tree partition number of an $r$-edge-colored graph $G$, denoted by $t_r(G)$, is the minimum integer $k$ such that whenever the edges of $G$ are colored with $r$ colors, the vertices of $G$ can be covered by at most $k$ vertex-disjoint monochromatic trees. In general, to determine this number is very difficult. For 2-edge-colored complete multipartite graph, Kaneko, Kano, and Suzuki gave the exact value of $t_2(K(n_1,n_2,...,n_k))$. In this paper, we prove that if $n\geq 3$, and K(n,n) is 3-edge-colored such that every vertex has color degree 3, then $t_3(K(n,n))=3.$