Spanning trees in the square of pseudorandom graphs

TL;DR

This paper proves that in certain pseudorandom graphs with high degree and bounded maximum degree, the square of the graph contains all spanning trees, answering a question in graph theory.

Contribution

It establishes a new condition under which the square of a pseudorandom graph contains all spanning trees with bounded degree, advancing understanding of graph powers.

Findings

Square of pseudorandom graphs contains all bounded degree spanning trees.

The result applies to graphs with sufficiently high degree and pseudorandomness.

Answers a previously open question by Krivelevich.

Abstract

We show that for every $\Delta\in\mathbb N$, there exists a constant $C$ such that if $G$ is an $(n,d,\lambda)$-graph with $d/\lambda\ge C$ and $d$ is large enough, then $G^2$ contains every $n$-vertex tree with maximum degree bounded by $\Delta$. This answers a question of Krivelevich.