A generalization of heterochromatic graphs

TL;DR

This paper introduces $f$-chromatic graphs, a generalization of heterochromatic graphs, and provides necessary and sufficient conditions for their spanning forests with specified components, extending previous results.

Contribution

It generalizes heterochromatic graphs to $f$-chromatic graphs and establishes new criteria for the existence of spanning forests with given component counts.

Findings

Provides a necessary and sufficient condition for $f$-chromatic spanning forests.

Shows that $g$-chromatic graphs with enough edges contain $f$-chromatic spanning forests.

Extends previous conditions for heterochromatic spanning trees to a broader class of graphs.

Abstract

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 $|E(G)|>\binom{n-m}{2}$ has an $f$-chromatic spanning forest with exactly $m$ ($1 \le m \le n-1$) components if $g(c) \le \frac{|E(G)|}{n-m}f(c)$ for any color $c$.