On 2-colored graphs and partitions of boxes

TL;DR

This paper proves a lower bound on the number of vertices in 2-colored graphs with certain monochromatic clique properties, confirming a conjecture and solving a specific case of a problem about partitioning discrete boxes.

Contribution

It establishes a minimum vertex count for such graphs and characterizes the equality case, confirming a conjecture and advancing understanding of box partitions.

Findings

Proved that G has at least 4(k-1) vertices under given conditions.

Confirmed the conjecture by Bucic et al. for the 2-dimensional case.

Characterized the equality case in the main theorem.

Abstract

We prove that if the edges of a graph G can be colored blue or red in such a way that every vertex belongs to a monochromatic k-clique of each color, then G has at least 4(k-1) vertices. This confirms a conjecture of Bucic, Lidicky, Long, and Wagner (arXiv:1805.11278[math.CO]) and thereby solves the 2-dimensional case of their problem about partitions of discrete boxes with the k-piercing property. We also characterize the case of equality in our result.