Equitable partition of planar graphs

Abstract

An equitable k-partition of a graph G is a collection of induced subgraphs (G[V-1], G[V-2], ... , G[V-k]) of G such that (V-1, V-2, ... , V-k) is a partition of V(G) and -1 <= |V-i| - |V-j| <= 1 for all 1 <= i < j <= k. We prove that every planar graph admits an equitable 2-partition into 3-degenerate graphs, an equitable 3-partition into 2-degenerate graphs, and an equitable 3-partition into two forests and one graph. (c) 2021 Elsevier B.V. All rights reserved.