Inverse Tur\'an numbers

TL;DR

This paper studies the inverse Turán number, a dual graph extremal problem, determining asymptotic values for paths of lengths 4 and 5, and providing bounds for even cycles, with conjectures on their asymptotic behavior.

Contribution

The paper determines asymptotic inverse Turán numbers for paths of length 4 and 5, improves bounds for even cycles, and proposes conjectures on their asymptotic behavior based on parity.

Findings

Asymptotic inverse Turán number for P4 and P5 determined.

Improved lower bounds for all even-length paths.

Enhanced bounds on inverse Turán numbers for C4.

Abstract

For given graphs $G$ and $F$, the Tur\'an number $ex(G,F)$ is defined to be the maximum number of edges in an $F$-free subgraph of $G$. Foucaud, Krivelevich and Perarnau and later independently Briggs and Cox introduced a dual version of this problem wherein for a given number $k$, one maximizes the number of edges in a host graph $G$ for which $ex(G,H) < k$. Addressing a problem of Briggs and Cox, we determine the asymptotic value of the inverse Tur\'an number of the paths of length $4$ and $5$ and provide an improved lower bound for all paths of even length. Moreover, we obtain bounds on the inverse Tur\'an number of even cycles giving improved bounds on the leading coefficient in the case of $C_4$. Finally, we give multiple conjectures concerning the asymptotic value of the inverse Tur\'an number of $C_4$ and $P_{\ell}$, suggesting that in the latter problem the asymptotic behavior depends heavily on the parity of $\ell$.