Graph Coloring and Function Simulation (1008.3015v1)

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