Graph Coloring and Function Simulation

We prove that every partial function with finite domain and range can be effectively simulated through sequential colorings of graphs. Namely, we show that given a finite set $S={0,1,ldots,m-1}$ and a number $n geq max{m,3}$, any partial function $varphi:S^{^p} to S^{^q}$ (i.e. it may not be defined on some elements of its domain $S^{^p}$) can be effectively (i.e. in polynomial time) transformed to a simple graph $matr{G}_{_{varphi,n}}$ along with three sets of specified vertices $$X = {x_{_{0}},x_{_{1}},ldots,x_{_{p-1}}}, Y = {y_{_{0}},y_{_{1}},ldots,y_{_{q-1}}}, R = {Kv{0},Kv{1},ldots,Kv{n-1}},$$ such that any assignment $sigma_{_{0}}: X cup R to {0,1,ldots,n-1} $ with $sigma_{_{0}}(Kv{i})=i$ for all $0 leq i < n$, is {it uniquely} and {it effectively} extendable to a proper $n$-coloring $sigma$ of $matr{G}_{_{varphi,n}}$ for which we have $$varphi(sigma(x_{_{0}}),sigma(x_{_{1}}),ldots,sigma(x_{_{p-1}}))=(sigma(y_{_{0}}),sigma(y_{_{1}}),ldots,sigma(y_{_{q-1}})),$$ unless $(sigma(x_{_{0}}),sigma(x_{_{1}}),ldots,sigma(x_{_{p-1}}))$ is not in the domain of $varphi$ (in which case $sigma_{_{0}}$ has no extension to a proper $n$-coloring of $matr{G}_{_{varphi,n}}$).