On the Packing Coloring Gap of Graphs

TL;DR

This paper introduces the packing coloring gap, measures how vertex deletion affects the packing chromatic number, and analyzes this parameter for trees, caterpillars, and other graphs.

Contribution

It defines the packing coloring gap and determines its values for specific graph classes like trees and caterpillars, extending to corona operations.

Findings

Packing coloring gap can be zero, one, or arbitrarily large.

Determined the packing coloring gap for caterpillars.

Extended results to caterpillars under the corona operation with K1.

Abstract

The packing chromatic number of a graph is the minimum number of colors for which the graph admits a packing coloring. This distance-based parameter may change under local structural modifications of the graph. In this paper, we introduce the packing coloring gap, defined as the maximum decrease in the packing chromatic number caused by the deletion of a single vertex. We focus on trees and determine the packing coloring gap for caterpillars. We further extend these results to caterpillars under the corona operation with K1. In addition, we present examples of graphs with packing coloring gap zero, one, and arbitrarily large.