Vertex Partitions into an Independent Set and a Forest with Each Component Small

TL;DR

This paper establishes sharp bounds on the maximum average degree for graph partitions into an independent set and a forest with small components, extending known results and answering a specific open question.

Contribution

It determines the exact maximum average degree bounds for such partitions for all relevant parameters, including new results for the case when one part is an independent set.

Findings

Sharp bounds on mad(G) for partitions into independent set and small-component forest.

Construction of infinite families showing bounds are tight.

Extension of known results to the case 4/3 < b < 2 for g(1,b).

Abstract

For each integer k >= 2, we determine a sharp bound on mad(G) such that V(G) can be partitioned into sets I and F_k, where I is an independent set and G[F_k] is a forest in which each component has at most k vertices. For each k we construct an infinite family of examples showing our result is best possible. Our results imply that every planar graph G of girth at least 9 (resp. 8, 7) has a partition of V(G) into an independent set I and a set F such that G[F] is a forest with each component of order at most 3 (resp. 4, 6). Hendrey, Norin, and Wood asked for the largest function g(a,b) such that if mad(G) < g(a,b) then V(G) has a partition into sets A and B such that mad(G[A]) < a and mad(G[B]) < b. They specifically asked for the value of g(1,b), i.e., the case when A is an independent set. Previously, the only values known were g(1,4/3) and g(1,2). We find g(1,b) whenever 4/3 < b < 2.