[Paper Review] The edge Folkman number $F_e(3, 3; 4)$ is greater than 19

Aleksandar Bikov, Nedyalko Nenov|arXiv (Cornell University)|Sep 12, 2016

Limits and Structures in Graph Theory13 references7 citations

TL;DR

This paper proves that the edge Folkman number $F_e(3,3;4)$ is at least 20, improving the previous lower bound of 19. Using a computer-assisted algorithm to exhaustively analyze maximal $K_4$-free graphs, the authors verify that no 19-vertex graph satisfies the Folkman condition, thereby establishing a new lower bound. The work also tightens bounds for the vertex Folkman number $F_v(2,3,3;4)$, showing it lies between 20 and 24.

ABSTRACT

The set of the graphs which do not contain the complete graph on $q$ vertices $K_q$ and have the property that in every coloring of their edges in two colors there exist a monochromatic triangle is denoted by $\mathcal{H}_e(3, 3; q)$. The edge Folkman numbers $F_e(3, 3; q) = \min\{|V(G)| : G \in \mathcal{H}_e(3, 3; q)\}$ are considered. Folkman proved in 1970 that $F_e(3, 3; q)$ exists if and only if $q \geq 4$. From the Ramsey number $R(3, 3) = 6$ it becomes clear that $F_e(3, 3; q) = 6$ if $q \geq 7$. It is also known that $F_e(3, 3; 6) = 8$ and $F_e(3, 3; 5) = 15$. The upper bounds on the number $F_e(3, 3; 4)$ which follow from the construction of Folkman and from the constructions of some other authors are not good. In 1975 Erdos posed the problem to prove the inequality $F_e(3, 3; 4) < 10^{10}$. This Erdos problem was solved by Spencer in 1978. The last upper bound on $F_e(3, 3; 4)$ was obtained in 2012 by Lange, Radziszowski and Xu, who proved that $F_e(3, 3; 4) \leq 786$. The best lower bound on this number is 19 and was obtained 10 years ago by Radziszowski and Xu. In this paper, we improve this result by proving $F_e(3, 3; 4) \geq 20$. At the end of the paper, we improve the known bounds on the vertex Folkman number $F_v(2, 3, 3; 4)$ by proving $20 \leq F_v(2, 3, 3; 4) \leq 24$.

Motivation & Objective

• To improve the lower bound on the edge Folkman number $F_e(3,3;4)$, which is the smallest number of vertices in a graph with no $K_4$ and every 2-edge-coloring containing a monochromatic triangle.
• To resolve an open problem in Ramsey theory concerning the existence and minimality of such graphs with bounded clique number.
• To extend the analysis to the vertex Folkman number $F_v(2,3,3;4)$, refining its known bounds.

Proposed method

• Developed a computer algorithm (Algorithm 2.4) to generate and analyze all maximal $K_4$-free graphs on 15 vertices with independence number at most 5.
• Used the property that any graph in $\mathcal{H}_e(3,3;4)$ must have chromatic number at least 6, and applied this to constrain candidate graphs.
• Executed a systematic search over all 2,081,234 graphs in $\mathcal{L}(15;1)$, identifying those not in $\mathcal{H}_v(3,3;4;15)$, and used them to test for $F_e(3,3;4)$-violating configurations.
• Applied Propositions 4.2 and 4.3 to restrict the structure of potential minimal graphs, particularly by analyzing vertex deletions and induced subgraphs.
• Verified that no 19-vertex graph satisfies the Folkman condition by checking all candidates in $\mathcal{B}$ after step 4 of Algorithm 2.4.
• Conducted a separate search over vertex-transitive graphs up to 31 vertices to identify those in $\mathcal{H}_v(2,3,3;4)$, leading to the upper bound of 24.

Experimental results

Research questions

• RQ1 Is there a 19-vertex graph with no $K_4$ such that every 2-edge-coloring contains a monochromatic triangle?
• RQ2 Can the lower bound on $F_e(3,3;4)$ be improved beyond 19 using computational enumeration?
• RQ3 What are the tightest known bounds for the vertex Folkman number $F_v(2,3,3;4)$?
• RQ4 Are there maximal $K_4$-free graphs on 19 vertices that satisfy the Folkman property for edge-colorings?
• RQ5 Can the structure of maximal $K_4$-free graphs be leveraged to prove non-existence of smaller Folkman graphs?

Key findings

• The edge Folkman number $F_e(3,3;4)$ is at least 20, improving the prior lower bound of 19.
• No 19-vertex graph exists in $\mathcal{H}_e(3,3;4)$, as confirmed by exhaustive search using Algorithm 2.4.
• The vertex Folkman number $F_v(2,3,3;4)$ satisfies $20 \leq F_v(2,3,3;4) \leq 24$, improving the previous bounds of $19 \leq F_v(2,3,3;4) \leq 30$.
• A 24-vertex vertex-transitive graph was found that belongs to $\mathcal{H}_v(2,3,3;4)$, and it is minimal in the sense that no proper subgraph satisfies the condition.
• The algorithm successfully identified all 2,081,234 maximal $K_4$-free graphs on 15 vertices with independence number at most 5, and verified that none of the 20 exceptional graphs in $\mathcal{L}(15;1) \setminus \mathcal{H}_v(3,3;5;15)$ satisfy the Folkman condition.
• The search confirmed that $\mathcal{H}_v(3,3;4;14) = \mathcal{L}(14;1)$, but $\mathcal{H}_v(3,3;4;15) \neq \mathcal{L}(15;1)$, highlighting structural differences in the graph space.