Vertex-minor Ramsey numbers: exact values and extremal structure

TL;DR

This paper determines the exact vertex-minor Ramsey number for four vertices, classifies extremal graphs avoiding this minor, and explores their structural properties and implications for larger numbers.

Contribution

It provides the first exact value of , classifies extremal graphs, and establishes new lower bounds for s growth, surpassing previous asymptotic bounds.

Findings

=11 is the exact vertex-minor Ramsey number.

Six extremal graphs on 10 vertices avoid  as a vertex-minor.

Lower bounds for s are improved, e.g., s lower bound is at least 13.

Abstract

We determine the vertex-minor Ramsey number $\Rvm(4)=11$, where $\Rvm(k)$ is the smallest~$n$ such that every $n$-vertex graph contains the edgeless graph~$E_k$ as a vertex-minor. We prove this by an exhaustive classification of the graphs on~$10$ and~$11$ vertices under local complementation. At the extremal order $n=10$, exactly six non-isomorphic graphs avoid~$E_4$ as a vertex-minor; up to isomorphism, they represent five LC-equivalence classes, and each labeled LC orbit has cardinality~$8{,}712$. Thus $k=4$ is the first case in which the general upper bound $2^k-1$ is not attained. Using the extremal graphs as building blocks, we derive explicit lower bounds on~$\Rvm(k)$ that surpass the leading term of the asymptotic bound for all $k\leq 9$; in particular, $\Rvm(5)\geq 13$. We also describe structural properties of the six extremal graphs and formulate the next open problem, whether $\Rvm(5)=15$.