On Small Folkman Graphs Arrowing $K_2$ or $K_3$

Abstract

For a graph $G$ and integers $a_i \geq 1$, we say that $G \xrightarrow[]{} (a_1, \ldots, a_k)^v$ if in any $k$-coloring of $G$'s vertices there exists a monochromatic $a_i$-clique for some color $i \in \{1,\ldots,k\}$. $G \xrightarrow[]{} (a_1, \ldots, a_k)^e$ is defined similarly, but for edge colorings. The Folkman number $F_v(a_1, \ldots, a_k; H)$ is the smallest number of vertices for which an $H$-free graph arrowing $(a_1, \ldots, a_k)^v$ exists. $F_e(a_1, \ldots, a_k; H)$ is defined similarly for edge-arrowing. In this work, we present new bounds for Folkman numbers where $a_i \in \{2,3\}$ and $k \leq 4$, while avoiding $K_n$, $J_n$, for $n \in \{4,5,6\}$, where $K_n$ is the complete graph on $n$ vertices and $J_n$ is $K_n$ missing an edge. We also present results for $C_4$-free and $W_5$-free graphs, where $C_4$ is the cycle on four vertices and $W_5$ is the wheel graph on five vertices. Notably, we prove the existence of $F_e(3,3;W_5)$, leaving only one graph, $\overline{P_2 \cup P_3}$, on five vertices for which the existence problem of $F_e(3,3;H)$ remains open. We provide some theoretical results that should aid in uncovering the existence of $F_v(3,3; \overline{P_2 \cup P_3})$. Our new bounds are the result of a variety of methods involving filters, extension, semi-polycirculant graphs, locally linear graphs, and the modification of special graphs. Most of our bounds are from the semi-polycirculant graph generator, showcasing its efficacy for finding witness Folkman graphs.