A heterochromatic tree is an edge-colored tree in which any two edges have different colors.The heterochromatic tree partition number of an r-edge-colored graph G,denoted by tr(G),is the minimum positive integer p such that whenever the edges of the graph G are colored with r colors,the vertices of G can be covered by at most p vertex-disjoint heterochromatic trees.In this paper we determine the heterochromatic tree partition number of r-edge-colored complete graphs.We also find at most tr(Kn) vertex-disjoint heterochromatic trees to cover all the vertices in polynomial time for a given r-edge-coloring of Kn.
Let G be a connected graph of order n. The rainbow connection number rc(G) of G was introduced by Chartrand et al. Chandran et al. used the minimum degree δ of G and obtained an upper bound that rc(G) ≤ 3n/(δ + 1) + 3, which is tight up to additive factors. In this paper, we use the minimum degree-sum σ2 of G to obtain a better bound rc(G) ≤6nσ2+2+ 8, especially when δ is small(constant) but σ2is large(linear in n).
