Abstract
In 2006, Suzuki, and Akbari and 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 \({|E(G)| > \binom{n-m}{2}}\) has an f-chromatic spanning forest with exactly m (1 ≤ m ≤ n − 1) components if \({g(c) \le \frac{|E(G)|}{n-m}f(c)}\) for any color c.
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This work was supported by MEXT. KAKENHI 21740085.
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Suzuki, K. A Generalization of Heterochromatic Graphs and f-Chromatic Spanning Forests. Graphs and Combinatorics 29, 715–727 (2013). https://doi.org/10.1007/s00373-011-1125-z
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DOI: https://doi.org/10.1007/s00373-011-1125-z
Keywords
- f-Chromatic
- Heterochromatic
- Rainbow
- Multicolored
- Totally multicolored
- Polychromatic
- Colorful
- Edge-coloring
- k-bounded coloring
- Spanning tree
- Spanning forest