Decomposing edge-colored graphs under color degree constraints

TL;DR

This paper proves a new partitioning theorem for edge-colored graphs with minimum color degree at least 5, linking it to a conjecture in digraph theory and advancing understanding of color degree constraints.

Contribution

The paper establishes a novel partitioning result for edge-colored graphs under color degree constraints, and connects it to Bermond-Thomassen's conjecture in digraphs.

Findings

Partition of vertices into two parts with minimum color degree at least 2

Strengthening of the main theorem with a more robust form

Connection between graph coloring and digraph conjecture

Abstract

For an edge-colored graph $G$, the minimum color degree of $G$ means the minimum number of colors on edges which are adjacent to each vertex of $G$. We prove that if $G$ is an edge-colored graph with minimum color degree at least $5$ then $V(G)$ can be partitioned into two parts such that each part induces a subgraph with minimum color degree at least $2$. We show this theorem by proving a much stronger form. Moreover, we point out an important relationship between our theorem and Bermond-Thomassen's conjecture in digraphs.