On Maximum Differential Coloring of Planar Graphs

TL;DR

This paper investigates the maximum differential coloring problem in graphs, proving its NP-hardness for planar graphs, establishing bounds for specific subclasses, and developing near-optimal algorithms for certain graph types.

Contribution

It proves NP-hardness for planar graphs, derives bounds for regular caterpillars and spiders, and proposes near-optimal algorithms for specific graph subclasses.

Findings

Maximum differential coloring is NP-hard for planar graphs.

Tight bounds are established for regular caterpillars and spider graphs.

Near-optimal algorithms are developed for certain subclasses like caterpillars and outer-planar graphs.

Abstract

We study the \emph{maximum differential coloring problem}, where the vertices of an $n$-vertex graph must be labeled with distinct numbers ranging from $1$ to $n$, so that the minimum absolute difference between two labels of any two adjacent vertices is maximized. As the problem is \NPH for general graphs~\cite{leung1984}, we consider planar graphs and subclasses thereof. We initially prove that the maximum differential coloring problem remains \NPH, even for planar graphs. Then, we present tight bounds for regular caterpillars and spider graphs. Using these new bounds, we prove that the Miller-Pritikin labeling scheme~\cite{miller89} for forests is optimal for regular caterpillars and for spider graphs. Finally, we describe close-to-optimal differential coloring algorithms for general caterpillars and biconnected triangle-free outer-planar graphs.