On path partitions of the divisor graph

TL;DR

This paper investigates how to partition the set of integers up to N into the fewest divisor graph paths, revealing that the longest path in such a partition can be nearly linear in N, contrasting with the longest simple path.

Contribution

It introduces a new perspective on divisor graph path partitions, showing that minimal partitions can contain very long paths, nearly linear in size, which was not previously known.

Findings

Longest path in minimal partitions is asymptotically N^{1-o(1)}

Longest simple path in divisor graph is about N / log N

Partitioning results differ from simple path length estimates

Abstract

It is known that the longest simple path in the divisor graph that uses integers $\leq N$ is of length $\asymp N/\log N$. We study the partitions of $\{1,2,\dots, N\}$ into a minimal number of paths of the divisor graph, and we show that in such a partition, the longest path can have length asymptotically $N^{1-o(1)}$.