In 2006, Suzuki, and Akbari & Alipour independently presented a necessary and
sufficient condition for edge-colored graphs to have a heterochromatic spanning
tree, where a heterochromatic spanning tree is a spanning tree whose edges have
distinct colors. In this paper, we propose f-chromatic graphs as a
generalization of heterochromatic graphs. An edge-colored graph is
f-chromatic if each color c appears on at most f(c) edges. We also
present a necessary and sufficient condition for edge-colored graphs to have an
f-chromatic spanning forest with exactly m components. Moreover, using this
criterion, we show that a g-chromatic graph G of order n with
β£E(G)β£>(2nβmβ) has an f-chromatic spanning forest with exactly m
(1β€mβ€nβ1) components if g(c)β€nβmβ£E(G)β£βf(c) for any
color c.Comment: 14 pages, 4 figure