Graph Coloring and Function Simulation

TL;DR

The paper demonstrates how any partial function with finite domain and range can be represented through graph colorings, enabling effective simulation of function evaluation via proper graph colorings.

Contribution

It introduces a polynomial-time method to encode partial functions as graphs such that their evaluation corresponds to extending partial colorings to proper colorings.

Findings

Partial functions can be effectively simulated by graph colorings.

The construction allows unique extension of initial colorings to proper colorings.

This approach bridges function computation and graph coloring problems.

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