Counting K_4-Subdivisions

TL;DR

This paper investigates the minimum number of $K_4$-subdivisions in 3-connected graphs, establishing a cubic lower bound, its tightness, and the computational hardness of counting these subdivisions.

Contribution

It proves a tight cubic lower bound on the number of $K_4$-subdivisions in 3-connected graphs and shows counting them is $ ext{ extsterling}P$-hard.

Findings

There are at least $oldsymbol{ ext{ extsterling}n^3}$ $K_4$-subdivisions in every 3-connected graph on $n$ vertices.

The cubic bound is tight for infinitely many graphs.

Counting the exact number of $K_4$-subdivisions is $ ext{ extsterling}P$-hard.

Abstract

A fundamental theorem in graph theory states that any 3-connected graph contains a subdivision of $K_4$. As a generalization, we ask for the minimum number of $K_4$-subdivisions that are contained in every $3$-connected graph on $n$ vertices. We prove that there are $\Omega(n^3)$ such $K_4$-subdivisions and show that the order of this bound is tight for infinitely many graphs. We further investigate a better bound in dependence on $m$ and prove that the computational complexity of the problem of counting the exact number of $K_4$-subdivisions is $\#P$-hard.