Splitting vertices of bipartite graphs preserves de Bruijn-Erdős property

TL;DR

The paper proves that every graph $G$ obtained from a bipartite graph by repeatedly splitting vertices into adjacent twins has the de Bruijn-Erdős property: either a universal line or at least $|V(G)|$ distinct lines. It introduces a blob-based decomposition, where blobs are the preimages of original vertices and are either rich or trivial. A key bound on the number of lines is $L \ge \binom{p}{2} + \binom{\lceil (n-p)/k \rceil}{2} + 2k$, with $p$ the number of vertices in rich blobs and $k$ the number of rich blobs with a trivial neighbor; the argument splits into three cases on $p$, supported by inequalities involving $\epsilon$ and a computer check for small $n$. Concluding, this establishes the conjectured property for the specified graph transformations.

Abstract

In this note, we prove that every graph obtained from a bipartite graph by iteratively splitting vertices into two adjacent twins has the de Bruijn-Erdős property.

Figures (1)

• Figure 1: Splitting vertices from some graph $H$ to some graph $G$.

Theorems & Definitions (9)

• Theorem 1
• Lemma 2
• proof
• Lemma 3: Theorem 6 in chvatal2018
• proof
• Claim 4
• Claim 5
• Claim 6
• Claim 7