Triangle-free extremal graph conjecture

Prove that, among all triangle-free graphs on n vertices, the complete bipartite graph with partite sets of sizes ceil(n/2) and floor(n/2) uniquely maximizes the subpath number.

Background

The paper proves that the complete bipartite graph with partite sets as equal in size as possible uniquely maximizes the subpath number among bipartite graphs on n vertices.

The authors propose extending this extremal result from bipartite graphs to the broader class of triangle-free graphs. This extension is not established in the paper and is explicitly stated as a conjecture.

References

Finally, based on our investigation of bipartite graphs, where the complete bipartite graph $K_{\left\lceil n/2\right\rceil ,\left\lfloor n/2\right\rfloor }$ maximizes the subpath number, we believe that the same graph maximizes the subpath number in a wider family of graphs also, namely the triangle-free graphs. We state it as the following conjecture. Among triangle-free graphs on $n$ vertices the maximum value of the subpath number is attained only by the complete bipartite graph $K_{\left\lceil n/2\right\rceil ,\left\lfloor n/2\right\rfloor }$.

— Invitation to the subpath number  (2503.00558 - Knor et al., 1 Mar 2025) in Section 6, Concluding remarks and further work, final conjecture