$(g,f)$-Chromatic spanning trees and forests

TL;DR

This paper introduces the concept of $(g,f)$-chromatic graphs, providing a necessary and sufficient condition for their spanning trees, and explores their properties in complete graphs with conjectures on partitioning into such trees.

Contribution

It defines $(g,f)$-chromatic graphs, establishes a criterion for their spanning trees, and proposes a conjecture on partitioning complete graphs into such trees with similar color distributions.

Findings

Established a necessary and sufficient condition for $(g,f)$-chromatic spanning trees.

Showed that complete graphs have spanning trees with similar color distributions.

Proposed a conjecture on partitioning complete graphs into edge-disjoint $(g,f)$-chromatic spanning trees.

Abstract

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 $2n$ $(n \ge 3)$ can be partitioned into $n$ edge-disjoint spanning trees such that each has a color probability distribution `similar' to that of $G$.