Tur\'an-Type Extremal Results for Distance-$k$ Graphs

TL;DR

This paper investigates extremal problems in distance graphs, determining maximum pairs at certain distances without triangles, and characterizes extremal graphs, advancing conjectures in the field.

Contribution

It resolves the first case of a conjecture on maximum distance-3 pairs and characterizes extremal graphs for distance-2 pairs without triangles.

Findings

Maximum pairs at distance three without triangles determined

Maximum pairs at distance two without triangles determined

Complete characterization of extremal graphs provided

Abstract

We study Tur\'an-type extremal problems for distance graphs, motivated by work of Csikv\'ari, Bollob\'as, Tyomkyn, and Uzzell. We determine the maximum number of vertex pairs at distance three in an $n$-vertex graph with no triangle formed by these pairs, resolving the first case of a conjecture of Tyomkyn and Uzzell. We also determine the maximum number of vertex pairs at distance two in an $n$-vertex graph with no triangle formed by these pairs and give a complete characterization of the extremal graphs, settling another problem of Tyomkyn and Uzzell.