On Simultaneous 2-locally-balanced 2-partition for Two Forests with Same Vertices

TL;DR

This paper proves the existence of a specific vertex partition for two forests with similar neighborhood capacities, demonstrating the limits of such partitions and providing an example to show the bounds are tight.

Contribution

It establishes the existence of a simultaneous 2-locally-balanced 2-partition for two forests with the same vertices and capacity constraints, and shows that the constant cannot be improved.

Findings

Partition exists under given capacity constraints

Improvement of the constant is impossible

Provides an example illustrating the bounds

Abstract

The existence of a partition of the common set of the vertices of two forests into two subsets, when difference of their capacities in the neighborhood of each vertex of each forest not greater than 2 is proved, and an example, which shows that improvement of the specified constant is impossible is brought.