Removing induced powers of cycles from a graph via fewest edits

TL;DR

This paper advances the understanding of the edit distance function for graphs avoiding induced cycles, specifically determining it for certain cycle lengths and improving bounds for even cycles.

Contribution

It fully determines the edit distance function for cycles of length 10 and 12, and extends results for even cycles and powers of cycles, improving previous bounds.

Findings

Determined $ ext{ed}_{ ext{Forb}(C_{10})}(p)$ and $ ext{ed}_{ ext{Forb}(C_{12})}(p)$.

Improved bounds for even cycles for all $p$ in a specified range.

Extended results to powers of cycles when $(t+1) ext{ divides } h$.

Abstract

What is the minimum proportion of edges which must be added to or removed from a graph of density $p$ to eliminate all induced cycles of length $h$? The maximum of this quantity over all graphs of density $p$ is measured by the edit distance function, $\text{ed}_{\text{Forb}(C_h)}(p)$, a function which provides a natural metric between graphs and hereditary properties. Martin determined $\text{ed}_{\text{Forb}(C_h)}(p)$ for all $p \in [0,1]$ when $h \in \{3, \ldots, 9\}$ and determined $\text{ed}_{\text{Forb}(C_{10})}(p)$ for $p \in [1/7, 1]$. Peck determined $\text{ed}_{\text{Forb}(C_h)}(p)$ for all $p \in [0,1]$ for odd cycles, and for $p \in [ 1/\lceil h/3 \rceil, 1]$ for even cycles. In this paper, we fully determine the edit distance function for $C_{10}$ and $C_{12}$. Furthermore, we improve on the result of Peck for even cycles, by determining $\text{ed}_{\text{Forb}(C_h)}(p)$ for all $p \in [p_0, 1/\lceil h/3 \rceil ]$, where $p_0 \leq c/h^2$ for a constant $c$. More generally, if $C_h^t$ is the $t$-th power of the cycle $C_h$, we determine $\text{ed}_{\text{Forb}(C_h^t)}(p)$ for all $p \geq p_0$ in the case when $(t+1) \mid h$, thus improving on earlier work of Berikkyzy, Martin and Peck.