Sparse Sets in Triangle-free Graphs

TL;DR

This paper investigates the size of the largest k-sparse sets in triangle-free graphs, establishing exact bounds for small graphs, analyzing growth rates, and exploring related Ramsey numbers using proof techniques and graph enumeration.

Contribution

It provides new bounds and exact values for largest k-sparse sets in small triangle-free graphs, and introduces methods to analyze their growth and related Ramsey numbers.

Findings

Triangle-free graphs of order 11 contain a 1-sparse 5-set

Order 13 graphs contain a 2-sparse 7-set

Order 8 graphs contain a 3-sparse 6-set

Abstract

A set of vertices is $k$-sparse if it induces a graph with a maximum degree of at most $k$. In this missive, we consider the order of the largest $k$-sparse set in a triangle-free graph of fixed order. We show, for example, that every triangle-free graph of order 11 contains a 1-sparse 5-set; every triangle-free graph of order 13 contains a 2-sparse 7-set; and every triangle-free graph of order 8 contains a 3-sparse 6-set. Further, these are all best possible. For fixed $k$, we consider the growth rate of the largest $k$-sparse set of a triangle-free graph of order $n$. Also, we consider Ramsey numbers of the following type. Given $i$, what is the smallest $n$ having the property that all triangle-free graphs of order $n$ contain a 4-cycle or a $k$-sparse set of order $i$. We use both direct proof techniques and an efficient graph enumeration algorithm to obtain several values for defective Ramsey numbers and a parameter related to largest sparse sets in triangle-free graphs, along with their extremal graphs.