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 $\matr{G}_{_{\varphi,n}}$).