On structural properties of some probable $R(3, 10)$-critical graphs

TL;DR

This paper investigates the structural properties of hypothetical $R(3,10)$-critical graphs, assuming the Ramsey number is 42, and characterizes their degree sequences, connectivity, and diameter.

Contribution

It provides a detailed structural analysis of $R(3,10)$-critical graphs, including possible degree sequences and properties assuming the Ramsey number is 42.

Findings

Minimum degree and vertex connectivity are equal, being 6, 7, or 8.

Diameter is either 2 or 3 for such graphs.

If diameter is 2 and minimum degree is 6, only 21 degree sequences are possible.

Abstract

The Ramsey number $R(s, t)$ is the smallest positive integer $n$ such that every graph on $n$ vertices contains either a clique of size $s$ or an independent set of size $t$. An $R(s,t)$-critical graph is a graph on $R(s,t)-1$ vertices that contains neither a clique of size $s$ nor an independent set of size $t$. It is known that $40\leq R(3, 10)\leq 42$. We study the structure of a $R(3,10)$-critical graphs by assuming $R(3, 10)=42$. We show that if such a graph exists then its minimum degree and vertex connectivity are the same and is $6, 7$ or $8$. Then we find all the possible degree sequences of such graphs. Further, we show that if such a graph exists, then its diameter is either $2$ or $3$, and if it has diameter $2$ and minimum degree $6$, then it has only $21$ choices for its degree sequence.