On the edit distance from $K_{2,t}$-free graphs II: Cases $t\geq 5$

TL;DR

This paper investigates the edit distance function for the hereditary property forbidding $K_{2,t}$ subgraphs when $t extgreater=5$, extending known results and identifying maximum values for odd $t$ using advanced graph constructions.

Contribution

The authors extend the interval of known edit distance function values for $orb(K_{2,t})$ when $t extgreater=5$ and determine the maximum for all odd $t$, employing novel techniques and constructions.

Findings

Extended the known interval of the edit distance function for $orb(K_{2,t})$

Determined the maximum value of the function for all odd $t$

Developed new graph constructions to improve upper bounds

Abstract

The edit distance between two graphs on the same vertex set is defined to be size of the symmetric difference of their edge sets. The edit distance function of a hereditary property, $\mathcal{H}$, is a function of $p$ and measures, asymptotically, the furthest graph with edge density $p$ from $\mathcal{H}$ under this metric. The edit distance function has proven to be difficult to compute for many hereditary properties. Some surprising connections to extremal graph theory problems, such as strongly regular graphs and the problem of Zarankiewicz, have been uncovered in attempts to compute various edit distance functions. In this paper, we address the hereditary property $\forb(K_{2,t})$ when $t\geq5$, the property of having no induced copy of the complete bipartite graph with 2 vertices in one class and $t$ in the other. This work continues from a prior paper by the authors. Employing an assortment of techniques and colored regularity graph constructions, we are able to extend the interval over which the edit distance function for this hereditary property is generally known and determine its maximum value for all odd $t$. We also explore several constructions to improve upon known upper bounds for the function.