On Convex Geometric Graphs with no $k+1$ Pairwise Disjoint Edges

TL;DR

This paper investigates the maximum edges in convex geometric graphs avoiding $k+1$ disjoint edges, characterizes extremal structures, and introduces new classes with specific boundary-independent sets, providing a complete solution for the maximum edge count under these constraints.

Contribution

The paper introduces a new class of extremal convex geometric graphs with boundary-independent sets, extending the understanding of maximum edges avoiding $k+1$ disjoint edges.

Findings

Existence of extremal graphs with boundary-independent sets of size $q$

Determination of the maximum number of edges $f(n,k,q)$ for all parameters

Optimal bounds for the size of independent sets on the boundary

Abstract

A well-known result of Kupitz from 1982 asserts that the maximal number of edges in a convex geometric graph (CGG) on $n$ vertices that does not contain $k+1$ pairwise disjoint edges is $kn$ (provided $n>2k$). For $k=1$ and $k=n/2-1$, the extremal examples are completely characterized. For all other values of $k$, the structure of the extremal examples is far from known: their total number is unknown, and only a few classes of examples were presented, that are almost symmetric, consisting roughly of the $kn$ "longest possible" edges of $CK(n)$, the complete CGG of order $n$. In order to understand further the structure of the extremal examples, we present a class of extremal examples that lie at the other end of the spectrum. Namely, we break the symmetry by requiring that, in addition, the graph admit an independent set that consists of $q$ consecutive vertices on the boundary of the convex hull. We show that such graphs exist as long as $q \leq n-2k$ and that this value of $q$ is optimal. We generalize our discussion to the following question: what is the maximal possible number $f(n,k,q)$ of edges in a CGG on $n$ vertices that does not contain $k+1$ pairwise disjoint edges, and, in addition, admits an independent set that consists of $q$ consecutive vertices on the boundary of the convex hull? We provide a complete answer to this question, determining $f(n,k,q)$ for all relevant values of $n,k$ and $q$.