Vertex Partitions into an Independent Set and a Forest with Each Component Small
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. 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), which corresponds to the case that A is an independent set. Previously, the only values known were g(1,4/3) and g(1,2). We find the value of g(1,b) whenever 4/3 < b < 2.
 Publication:

arXiv eprints
 Pub Date:
 June 2020
 arXiv:
 arXiv:2006.11445
 Bibcode:
 2020arXiv200611445C
 Keywords:

 Mathematics  Combinatorics
 EPrint:
 21 pages, 9 figures and 2 tables, comments welcome