Threshold-Coloring and Unit-Cube Contact Representation of Graphs

TL;DR

This paper explores threshold coloring of graphs, identifying subclasses that are colorable, and connects these results to unit-cube contact representations, while analyzing computational complexity of related variants.

Contribution

It characterizes subclasses of graphs that are threshold-colorable and establishes links to contact representations and complexity results for related problems.

Findings

Certain subclasses of planar graphs are threshold-colorable.

Threshold coloring enables unit-cube contact representations for some graph classes.

Some variants of threshold coloring are NP-complete, while others are polynomial-time solvable.

Abstract

In this paper we study threshold coloring of graphs, where the vertex colors represented by integers are used to describe any spanning subgraph of the given graph as follows. Pairs of vertices with near colors imply the edge between them is present and pairs of vertices with far colors imply the edge is absent. Not all planar graphs are threshold-colorable, but several subclasses, such as trees, some planar grids, and planar graphs without short cycles can always be threshold-colored. Using these results we obtain unit-cube contact representation of several subclasses of planar graphs. Variants of the threshold coloring problem are related to well-known graph coloring and other graph-theoretic problems. Using these relations we show the NP-completeness for two of these variants, and describe a polynomial-time algorithm for another.