Chromatic Vertex Folkman Numbers

For graph $G$ and integers $a_1 ge cdots ge a_r ge 2$, we write $G rightarrow (a_1 ,cdots ,a_r)^v$ if and only if for every $r$-coloring of the vertex set $V(G)$ there exists a monochromatic $K_{a_i}$ in $G$ for some color $i in {1, cdots, r}$. The vertex Folkman number $F_v(a_1 ,cdots ,a_r; s)$ is defined as the smallest integer $n$ for which there exists a $K_s$-free graph $G$ of order $n$ such that $G rightarrow (a_1 ,cdots ,a_r)^v$. It is well known that if $G rightarrow (a_1 ,cdots ,a_r)^v$ then $chi(G) geq m$, where $m = 1+ sum_{i=1}^r (a_i - 1)$. In this paper we study such Folkman graphs $G$ with chromatic number $chi(G)=m$, which leads to a new concept of chromatic Folkman numbers. We prove constructively some existential results, among others that for all $r,s ge 2$ there exist $K_{s+1}$-free graphs $G$ such that $G rightarrow (s,cdots_r,s)^v$ and $G$ has the smallest possible chromatic number $r(s-1)+1$ for this $r$-color arrowing to hold. We also conjecture that, in some cases, our construction is the best possible, in particular that for every $s ge 2$ there exists a $K_{s+1}$-free graph $G$ on $F_v(s,s; s+1)$ vertices with $chi(G)=2s-1$ such that $G rightarrow (s,s)^v$.