Partitions of Graphs into Special Bipartite Graphs

TL;DR

The main result of this paper is the proof of the bounds for $χ'_{2K_2}(n)$, which corresponds to the minimum number of induced bipartite subgraphs needed to partition the edges of $K_n$.

Abstract

We study the problem of partitioning the edge set of the complete graph into bipartite subgraphs under certain constraints defined by forbidden subgraphs. These constraints lead to both classical problems, such as partitioning into independent matchings or complete bipartite subgraphs, and novel variants motivated by structural restrictions. Our theoretical framework is inspired by clustering problems in real-world transaction graphs, which can be formulated naturally as edge partitioning problems under bipartite graph constraints. The main result of this paper is the proof of the bounds for $χ'_{2K_2}(n)$, which corresponds to the minimum number of induced $2K_2$-free bipartite subgraphs needed to partition the edges of $K_n$. In addition to this central result, we also present several similar bounds for other forbidden subgraphs on three or four vertices. Some are included primarily for the sake of completeness, to demonstrate the broad applicability of our approach, and some lead to other novel or well-known graph theoretical problems.

Figures (8)

• Figure 1: Small excluded subgraphs
• Figure 2: Double star partition of $K_6$.
• Figure 3: Cherry Orchard.
• Figure 4: A descent edge $e$ and an ascent edge $f$ in $K_{25}$
• Figure 5: Illustration of $G_i$ and $G'_j$ (for $n=16$)

Theorems & Definitions (18)

• Definition 1
• Definition 2
• proof
• Remark
• Theorem 5: Graham--Pollak GP
• Theorem 8
• Theorem 9
• Remark
• Theorem 11
• Theorem 12: Cherry Orchard