13 research outputs found
Equitable partition of graphs into induced forests
An equitable partition of a graph is a partition of the vertex-set of
such that the sizes of any two parts differ by at most one. We show that every
graph with an acyclic coloring with at most colors can be equitably
partitioned into induced forests. We also prove that for any integers
and , any -degenerate graph can be equitably
partitioned into induced forests.
Each of these results implies the existence of a constant such that for
any , any planar graph has an equitable partition into induced
forests. This was conjectured by Wu, Zhang, and Li in 2013.Comment: 4 pages, final versio
Equitable partition of planar graphs
An equitable -partition of a graph is a collection of induced
subgraphs of such that
is a partition of and
for all . We prove that every planar graph admits an equitable
-partition into -degenerate graphs, an equitable -partition into
-degenerate graphs, and an equitable -partition into two forests and one
graph.Comment: 12 pages; revised; accepted to Discrete Mat