Density of 4-edge paths in graphs with fixed edge density

TL;DR

This paper determines the maximum number of 4-edge paths in graphs with fixed vertices and edges, identifying extremal structures and providing bounds, extending classical and recent results in graph theory.

Contribution

It offers an asymptotically sharp upper bound for 4-edge paths and characterizes extremal graphs based on edge density, extending known theorems to longer paths.

Findings

Asymptotically sharp upper bound for 4-edge paths

Extremal graphs are quasi-star or quasi-clique

Provides an easy lower bound for the number of 4-edge paths

Abstract

We investigate the number of 4-edge paths in graphs with a fixed number of vertices and edges. An asymptotically sharp upper bound is given to this quantity. The extremal construction is the quasi-star or the quasi-clique graph, depending on the edge density. An easy lower bound is also proved. This answer resembles the classic theorem of Ahlswede and Katona about the maximal number of 2-edge paths, and a recent theorem of Kenyon, Radin, Ren and Sadun about k-edge stars.