Structure of fixed-edge extremal graphs

Prove that every graph \(G\) on \(n\) vertices and \(m\) edges satisfying \(\beta_1(X(G))=\max_{H\in\mathbf{G}(n,m)}\beta_1(X(H))\) contains a complete bipartite spanning subgraph.

Background

The paper points out that extremal first Betti numbers for a fixed number of vertices and edges need not be attained by graphs containing the balanced Turán graph as a spanning subgraph. It gives an example with eight vertices and 21 edges in which a graph based on a different vertex partition has a larger first Betti number than graphs containing T8,2\mathcal{T}_{8,2}. The proposed conjecture seeks a broader structural characterization: despite this failure of balanced-Turán containment, every fixed-edge extremizer should contain some complete bipartite spanning subgraph.

References

Finding extremal values of \mathbb{Z}-linear functions defined on simplicial complexes with precisely n vertices and m edges presents a challenging and interesting direction for future research. Let \mathbf{G}(n,m) denote the collection of graphs on n vertices and m edges. If G \in \mathbf{G}(n,m) and \beta_1(G) = \max_{H \in \mathbf{G}(n,m)} \beta_1(H), then G contains a complete bipartite spanning subgraph.

— Extremal Betti Numbers and Persistence in Flag Complexes  (2502.21294 - Beers et al., 28 Feb 2025) in Section 6, Discussion