(g,f)-Chromatic spanning trees and forests
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A heterochromatic (or rainbow) graph is an edge-colored graph whose edges have distinct colors, that is, where each color appears at most once. In this paper, I propose a (g,f)-chromatic graph as an edge-colored graph where each color c appears at least g(c) times and at most f(c) times. I also present a necessary and sufficient condition for edge-colored graphs (not necessary to be proper) to have a (g,f)-chromatic spanning tree. Using this criterion, I show that an edge-colored complete graph G has a spanning tree with a color probability distribution `similar' to that of G. Moreover, I conjecture that an edge-colored complete graph G of order 2n (n > 3) can be partitioned into n edge-disjoint spanning trees such that each has a color probability distribution `similar' to that of G.
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