On Sequential Coloring of Graphs and its Defining Sets

TL;DR

This paper extends the concept of sequential graph coloring by introducing rule-based frameworks and defining new parameters, analyzing their properties and computational complexity, with some problems proven NP-complete.

Contribution

It generalizes sequential coloring to rule-based frameworks, introduces weak and strong defining numbers, and studies their properties and NP-completeness.

Findings

Weak and strong defining numbers are fundamentally different.

The spectra of these parameters are nontrivial.

Computing these parameters is NP-complete in many cases.

Abstract

In this paper, based on the contributions of Tucker (1983) and Seb{\H{o}} (1992), we generalize the concept of a sequential coloring of a graph to a framework in which the algorithm may use a coloring rule-base obtained from suitable forcing structures. In this regard, we introduce the {\it weak} and {\it strong sequential defining numbers} for such colorings and as the main results, after proving some basic properties, we show that these two parameters are intrinsically different and their spectra are nontrivial. Also, we consider the natural problems related to the complexity of computing such parameters and we show that in a variety of cases these problems are ${\bf NP}$-complete. We conjecture that this result does not depend on the rule-base for all nontrivial cases.