Local conditions for exponentially many subdivisions

TL;DR

This paper characterizes graphs for which the number of subdivisions guaranteed under local edge conditions grows exponentially with respect to a parameter, showing only complete graphs exhibit such exponential growth.

Contribution

It proves that only complete graphs have exponentially many subdivisions under local conditions, and characterizes graphs with exponential subdivision growth under strengthened conditions.

Findings

Only complete graphs have exponential subdivision counts under local conditions.

Non-complete graphs have polynomial subdivision counts in parameter t.

Characterization of graphs with exponential subdivision growth under strengthened conditions.

Abstract

Given a graph $F$, let $s_t(F)$ be the number of subdivisions of $F$, each with a different vertex set, which one can guarantee in a graph $G$ in which every edge lies in at least $t$ copies of $F$. In 1990, Tuza asked for which graphs $F$ and large $t$, one has that $s_t(F)$ is exponential in a power of $t$. We show that, somewhat surprisingly, the only such $F$ are complete graphs, and for every $F$ which is not complete, $s_t(F)$ is polynomial in $t$. Further, for a natural strengthening of the local condition above, we also characterise those $F$ for which $s_t(F)$ is exponential in a power of $t$.