Near-colorings: non-colorable graphs and NP-completeness

TL;DR

This paper investigates the computational complexity of near-colorings in graphs, establishing NP-completeness results for certain classes of planar graphs with specified girth and degree constraints, and providing explicit non-colorable examples.

Contribution

It proves NP-completeness for near-coloring problems in planar graphs with girth constraints and introduces explicit non-colorable planar graph examples.

Findings

NP-completeness of (k,j)-colorability for planar graphs with girth at least g

NP-completeness of determining (0,0,0)-colorability in certain planar graphs

Explicit examples of non-(3,1)-colorable and non-(2,0)-colorable planar graphs

Abstract

A graph G is (d_1,..,d_l)-colorable if the vertex set of G can be partitioned into subsets V_1,..,V_l such that the graph G[V_i] induced by the vertices of V_i has maximum degree at most d_i for all 1 <= i <= l. In this paper, we focus on complexity aspects of such colorings when l=2,3. More precisely, we prove that, for any fixed integers k,j,g with (k,j) distinct form (0,0) and g >= 3, either every planar graph with girth at least g is (k,j)-colorable or it is NP-complete to determine whether a planar graph with girth at least g is (k,j)-colorable. Also, for any fixed integer k, it is NP-complete to determine whether a planar graph that is either (0,0,0)-colorable or non-(k,k,1)-colorable is (0,0,0)-colorable. Additionally, we exhibit non-(3,1)-colorable planar graphs with girth 5 and non-(2,0)-colorable planar graphs with girth 7.