Partitioning Sparse Graphs into an Independent Set and a Forest of Bounded Degree

François Dross,Alexandre Pinlou,Mickael Montassier

doi:10.37236/6815

Abstract

An $({\cal I},{\cal F}_d)$-partition of a graph is a partition of the vertices of the graph into two sets $I$ and $F$, such that $I$ is an independent set and $F$ induces a forest of maximum degree at most $d$. We show that for all $M<3$ and $d \ge \frac{2}{3-M} - 2$, if a graph has maximum average degree less than $M$, then it has an $({\cal I},{\cal F}_d)$-partition. Additionally, we prove that for all $\frac{8}{3} \le M < 3$ and $d \ge \frac{1}{3-M}$, if a graph has maximum average degree less than $M$ then it has an $({\cal I},{\cal F}_d)$-partition. It follows that planar graphs with girth at least $7$ (resp. $8$, $10$) admit an $({\cal I},{\cal F}_5)$-partition (resp. $({\cal I},{\cal F}_3)$-partition, $({\cal I},{\cal F}_2)$-partition).

Highlights

In this paper, unless we specify otherwise, all the graph considered are simple graphs, without loops or multi-edges
For i classes of graphs G1, . . . , Gi, a (G1, . . . , Gi)-partition of a graph G is a partition of the vertices of G into i sets V1, . . . , Vi such that, for all 1 ≤ j ≤ i, the graph G[Vj] induced by Vj belongs to Gj
An (I, F, ∆2)-partition of G is a vertex-partition into three sets V1, V2, V3 such that G[V1] is an empty graph, G[V2] is a forest, and G[V3] is a graph with maximum degree at most 2

Summary

Introduction

Unless we specify otherwise, all the graph considered are simple graphs, without loops or multi-edges. Proved that every planar graph admits an (I, F , F )-partition, and Borodin and Glebov [4]. Proved that every planar graph with girth at least 5 admits an (I, F )-partition. Proved that every planar graph admits an (F2, F2, F2)-partition. 12 5 admits an (I, ∆1)-partition, which implies that that every planar graph with girth at least 12 admits an (I, ∆1)-partition. For every d, there exists a planar graph of girth at least 6 that admits no (I, ∆d)partition;. Every planar graph with girth at least 11 admits an (I, F1)-partition.

Proof of Theorem 1

Discharging procedure

Proof of Theorem 2