On structural properties of some probable $R(3, 10)$-critical graphs
Dinesh Pandey, Peruvemba Sundaram Ravi
TL;DR
The paper investigates $R(s,t)$-critical graphs for the case $R(3,10)$, alternating between a noncomputational study under the conjecture $R(3,10)=42$ and a preparatory discussion for the actual bound $R(3,10)=41$. It develops a structural framework for $(3,10,41)$ graphs, proving $Δ(Γ)=9$, $δ(Γ)∈{6,7,8}$, and $κ(Γ)=δ(Γ)$, and it exhaustively catalogs possible degree sequences, with a detailed diameter analysis showing $diam(Γ)∈{2,3}$ and a tight 21-sequence bound in the diameter-2, $δ=6$ scenario. The work extends to $(3,10,40)$ graphs under the alternative bound $R(3,10)=41$, deriving similar degree, diameter, and connectivity properties, and it develops intricate lemmas to bound neighbor-sets and degree distributions. Overall, the results provide significant structural constraints that could support noncomputational proofs and inform subsequent computational approaches to bounding $R(3,10)$ and related Ramsey numbers.
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.