Decomposing planar graphs into graphs with degree restrictions

TL;DR

This paper investigates how to partition planar graphs into two parts with specific degree restrictions, establishing exact and bounded values for the minimum degree parameters needed for such decompositions.

Contribution

It determines the exact value of $h_4$ and $h_3$, and bounds for $h_2$, advancing understanding of graph decompositions with degree constraints.

Findings

$h_4=1$ for all planar graphs

$h_3=2$ for all planar graphs

$4 \,\leq\, h_2 \leq 6$ for all planar graphs

Abstract

Given a graph $G$, a decomposition of $G$ is a partition of its edges. A graph is $(d, h)$-decomposable if its edge set can be partitioned into a $d$-degenerate graph and a graph with maximum degree at most $h$. For $d \le 4$, we are interested in the minimum integer $h_d$ such that every planar graph is $(d,h_d)$-decomposable. It was known that $h_3 \le 4$, $h_2\le 8$, and $h_1 = \infty$. This paper proves that $h_4=1, h_3=2$, and $4 \le h_2 \le 6$.