Edge partitions into induced-$2K_2$-free bipartite graphs

TL;DR

This paper investigates how to partition bipartite graphs into induced-$2K_2$-free bipartite graphs, providing bounds, exact values for specific graphs, and exploring the relationship with various graph parameters.

Contribution

It introduces bounds and exact computations for the minimum number of parts needed to partition bipartite graphs into Ferrers graphs, advancing understanding of their structural properties.

Findings

Computed $p(G)$ for paths and even cycles.

Established bounds based on induced matchings and Dilworth widths.

Identified separations indicating bounds can be far from tight.

Abstract

We study edge partitions of a bipartite graph into induced-$2K_2$-free bipartite graphs, i.e.\ into Ferrers (chain) graphs. We define $\fp(G)$ as the minimum number of parts in such a partition. We prove general lower and upper bounds in terms of induced matchings and Dilworth widths of neighborhood posets. We compute the parameter exactly for paths and even cycles, and we exhibit separations showing that the induced-matching lower bound and the width upper bound can both be far from tight. We also record a simple host-induced conflict-graph lower bound, present a $0$--$1$ matrix viewpoint, and add some complexity remarks.