[Paper Review] Modified vertex Folkman numbers

Aleksandar Bikov, Nedyalko Nenov|arXiv (Cornell University)|Nov 6, 2015

Limits and Structures in Graph Theory11 references3 citations

TL;DR

This paper establishes the exact value of the modified vertex Folkman number $Ϫ{F}({m}\big|_{6};m-1) = m+10$ for all $m \geq 8$, using a computational approach based on exhaustive search and graph generation algorithms. The authors prove that no graph of order $m+10$ satisfies the Folkman condition while having clique number less than $m-1$, resolving the case $p=6$ in the modified vertex Folkman number framework.

ABSTRACT

Let $a_1, ..., a_s$ be positive integers. For a graph $G$ the expression $$ G \overset{v}{ ightarrow} (a_1, ..., a_s) $$ means that for every coloring of the vertices of $G$ in $s$ colors ($s$-coloring) there exists $i \in \{1, ..., s\}$, such that there is a monochromatic $a_i$-clique of color $i$. If $m$ and $p$ are positive integers, then $$ G \overset{v}{ ightarrow} {m}\big\vert_{p} $$ means that for arbitrary positive integers $a_1, ..., a_s$ ($s$ is not fixed), such that $\sum_{i = 1}^{s}(a_i - 1) + 1 = m$ an $\max{\{a_1, ..., a_s\}} \leq p$ we have $G \overset{v}{ ightarrow} (a_1, ..., a_s)$. Let $$ \widetilde{\mathcal{H}}({m}\big\vert_{p}; q) = \{G : G \overset{v}{ ightarrow} {m}\big\vert_{p} \mbox{ and } ω(G) < q\}. $$ The modified vertex Folkman numbers are defined by the equality $$ \widetilde{F}({m}\big\vert_{p}; q) = \min{\{|V(G)| : G \in \widetilde{\mathcal{H}}({m}\big\vert_{p}; q)\}}. $$ If $q \geq m$ these numbers are known and they are easy to compute. In the case $q = m - 1$ we know all of the numbers when $p \leq 5$. In this work we consider the next unknown case $p = 6$ and we prove with the help of a computer that $$ \widetilde{F}({m}\big\vert_{6}; m - 1) = m + 10. $$

Motivation & Objective

• To determine the exact value of the modified vertex Folkman number $Ϫ{F}({m}\big|_{6};m-1)$ for $m \geq 8$, extending known results for $p \leq 5$.
• To resolve the previously unknown case of $p=6$ in the modified vertex Folkman number framework, which involves bounding the order of graphs that force monochromatic cliques under arbitrary $s$-colorings with bounded clique size.
• To establish that no graph of order $m+10$ exists satisfying the Folkman condition with clique number less than $m-1$, proving the minimality of the bound.

Proposed method

• The authors define the modified vertex Folkman number $Ϫ{F}({m}\big|_{p};q)$ as the minimal order of a graph $G$ such that $G \overset{v}{\rightarrow} m\big|_p$ and $\omega(G) < q$, with $m = \sum (a_i - 1) + 1$ and $\max a_i \leq p$.
• They employ two specialized algorithms: Algorithm 3.2 for generating maximal graphs with independence number greater than 2, and Algorithm 3.4 for generating graphs with independence number exactly 2, both starting from base complete graphs and adding independent vertices.
• The search is conducted recursively through nested sets $\widetilde{\mathcal{H}}({m}\big|_{6};{m-1};{m+9})$, verifying the non-existence of maximal graphs in these sets for $m = 9, 10, 11$.
• The proof relies on verifying that no such graphs exist in $\widetilde{\mathcal{H}}({m}\big|_{6};{m-1};{m+9})$ for $m = 9, 10, 11$, which implies $\widetilde{F}({m}\big|_{6};m-1) > m+9$, thus proving $\widetilde{F}({m}\big|_{6};m-1) = m+10$.
• The computation is validated by checking the non-existence of maximal graphs in all relevant sets, using independence number constraints and graph extension techniques.

Experimental results

Research questions

• RQ1 What is the exact value of the modified vertex Folkman number $\widetilde{F}({m}\big|_{6};m-1)$ for $m \geq 8$?
• RQ2 Does there exist a graph of order $m+10$ that satisfies $G \overset{v}{\rightarrow} m\big|_6$ while having clique number less than $m-1$?
• RQ3 Can the non-existence of such graphs be computationally verified for $m = 9, 10, 11$ using recursive graph generation and independence number constraints?

Key findings

• The modified vertex Folkman number $\widetilde{F}({m}\big|_{6};m-1)$ is exactly $m+10$ for all $m \geq 8$, resolving the $p=6$ case.
• No maximal graphs exist in the set $\widetilde{\mathcal{H}}({8}\big|_{6};{7};{17})$, confirming $\widetilde{F}({8}\big|_{6};{7}) = 18$.
• The non-existence of maximal graphs in $\widetilde{\mathcal{H}}({m}\big|_{6};{m-1};{m+9})$ for $m = 9, 10, 11$ implies $\widetilde{F}({m}\big|_{6};m-1) > m+9$, thus proving the exact value is $m+10$.
• The proof is computationally verified using two algorithms: one for graphs with independence number greater than 2 and another for those with independence number exactly 2, both applied recursively through nested graph sets.
• The result extends previous knowledge, which was only known for $p \leq 5$, and establishes a new exact value for the next open case in the modified vertex Folkman number sequence.