An f-chromatic spanning forest of edge-colored complete bipartite graphs

In 2001, Brualdi and Hollingsworth proved that an edge-colored balanced complete bipartite graph Kn,n with a color set C = {1, 2, 3, . . . , 2n − 1} has a heterochromatic spanning tree if the number of edges colored with colors in R is more than |R|/4 for any non-empty subset R ⊆ C, where a heterochromatic spanning tree is a spanning tree whose edges have distinct colors, namely, any color appears at most once. In 2010, Suzuki generalized heterochromatic graphs to f chromatic graphs, where any color c appears at most f(c). Moreover, he presented a necessary and sufficient condition for graphs to have an f -chromatic spanning forest with exactly w components. In this paper, using this necessary and sufficient condition, we generalize the Brualdi-Hollingsworth theorem above. Keyword(s): f -chromatic, heterochromatic, rainbow, multicolored, totally multicolored, polychromatic, colorful, edge-coloring, spanning tree, spanning forest. MSC2010: 05C05, 05C15.