The Tur\'an number of the square of a path

TL;DR

This paper determines the exact maximum number of edges in large graphs avoiding the square of paths with 5 and 6 vertices, advancing understanding of Turán numbers for these specific graph structures.

Contribution

It provides exact Turán numbers for the square of paths with 5 and 6 vertices and proposes a conjecture for larger cases.

Findings

Exact Turán number for P^2_5 established

Exact Turán number for P^2_6 established

Conjecture for general P^2_k proposed

Abstract

The Tur\'an number of a graph H, ex(n,H), is the maximum number of edges in a graph on n vertices which does not have H as a subgraph. Let P_k be the path with k vertices, the square P^2_k of P_k is obtained by joining the pairs of vertices with distance one or two in P_k. The powerful theorem of Erd\H{o}s, Stone and Simonovits determines the asymptotic behavior of ex(n,P^2_k). In the present paper, we determine the exact value of ex(n,P^2_5) and ex(n,P^2_6) and pose a conjecture for the exact value of ex(n,P^2_k).