Bounds for some generalized vertex Folkman numbers

Abstract

For a graph $G$ and positive integers $a_1,\dots ,a_r,$ if every r-coloring of vertics V(G) must result in a monochromatic $a_1$-clique of color $i$ for some $i\in \{1,\dots ,r\},$ then we write $G\to (a_1,..a_r{)}^v$.$F_v(K_{a}1,\dots ,K_{a}r;H)$ is the smallest integer $n$ such that there is an H-free graph $G$ of order $n$, and $G\to (a_1,\dots ,a_r{)}^v$. In this paper we study upper and lower bounds for some generalized vertex Folkman numbers of from $F_v(K_{a1},\dots ,K_{ar};K_4–e)$, where $r\in 2,3$ and $a_1\in 2,3$ for 10 and $F_v(K_2,K_3;K_4–e)=19$ by computing, and prove $F_v(K_3,K_3;K_4–e)\ge F_v(K_2,K_2,K_3;K_4–e)\ge 25$