A generalization of heterochromatic graphs

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 |E(G)| > ( n−m 2 ) has an f -chromatic spanning forest with exactly m (1 ≤ m ≤ n − 1) components if g(c) ≤ |E(G)| n−m f(c) for any color c. Keyword(s): f -chromatic, heterochromatic, rainbow, multicolored, totally multicolored, polychromatic, colorful, edge-coloring, k-bounded coloring, spanning tree, spanning forest. MSC2010: 05C05, 05C15.