Paths are Tur\'an-good

TL;DR

This paper proves that among graphs without a complete subgraph of size k+1, the Turán graph maximizes the number of paths, establishing a Turán-type extremal property for paths.

Contribution

It establishes that Turán graphs are extremal for maximizing path counts in K_{k+1}-free graphs, extending Turán's theorem to path enumeration.

Findings

Turán graph contains the most paths among K_{k+1}-free graphs.

The result generalizes Turán's theorem to path counts.

Provides a new extremal characterization for paths in forbidden clique graphs.

Abstract

We show that among $K_{k+1}$-free $n$-vertex graphs, the Tur\'an graph contains the most copies of any path.