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

TL;DR

This paper generalizes a theorem about heterochromatic spanning trees in edge-colored bipartite graphs to f-chromatic spanning forests, providing a broader framework for understanding color constraints in such graphs.

Contribution

It extends the Brualdi-Hollingsworth theorem to f-chromatic spanning forests, offering a necessary and sufficient condition for their existence in bipartite graphs.

Findings

Generalized heterochromatic spanning tree conditions to f-chromatic forests

Provided a necessary and sufficient condition for f-chromatic spanning forests

Extended previous results to graphs with multiple components

Abstract

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|^2 /4 for any non-empty subset R \subseteq 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.