Partitioning by Monochromatic Trees

Partitioning by Monochromatic Trees

Partitioning complete multipartite graphs by monochromatic trees

The tree partition number of an r-edge-colored graph G, denoted by tr(G), is the minimum number 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. We determine t2(K(n1, n2, . . . , nk)) of the complete k-partite graph K(n1, n2, . . . , nk). In particular, we prove that t2(K(n,m)) = (m − 2)/2n + 2, whe...

The complexity for partitioning graphs by monochromatic trees, cycles and paths

Let G be an edge-colored graph. We show in this paper that it is NP-hard to find the minimum number of vertex disjoint monochromatic trees which cover the vertices of the graph G. We also show that there is no constant factor approximation algorithm for the problem unless P = NP. The same results hold for the problem of finding the minimum number of vertex disjoint monochromatic cycles (paths, ...

Partitioning 3-Edge-Colored Complete Equi-Bipartite Graphs by Monochromatic Trees under a Color Degree Condition

The monochromatic tree partition number of an r-edge-colored graph G, denoted by tr(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 2edge-colored complete multipartite graph, Kaneko, Kano, and Suzuki gave the exa...

Vertex coverings by monochromatic cycles and trees

Assume that K is a finite complete graph whose edges are colored with r colors (r ~ 2 ). How many monochromatic paths (or cycles) are needed to cover (or partition) the vertex set of K? Throughout the paper single vertices and edges are considered to be cycles. It is not obvious that these numbers depend only on r. The following conjecture is from [ 12]. If the edges of a (finite undirected) co...