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تسجيل مستخدم جديدEstimating quantum chromatic numbers
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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 v
ertex 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 $G rightarrow (s,s)^v$.
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The total dominator total coloring of a graph is a total coloring of the graph such that each object of the graph is adjacent or incident to every object of some color class. The minimum namber of the color classes of a total dominator total coloring
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