Characterization of cycle obstruction sets for improper coloring planar graphs

TL;DR

This paper characterizes all minimal cycle obstruction sets that determine when planar graphs can be partitioned into parts with bounded degrees, generalizing improper coloring conditions for planar graphs.

Contribution

It provides a complete characterization of minimal cycle obstruction sets for balanced and unbalanced improper colorings of planar graphs for all values of k.

Findings

Identifies all minimal cycle obstruction sets for balanced k-partitionability.

Identifies all minimal cycle obstruction sets for unbalanced k-partitionability.

Provides a unified framework for understanding cycle obstructions in improper coloring of planar graphs.

Abstract

For nonnegative integers $k, d_1, \ldots, d_k$, a graph is $(d_1, \ldots, d_k)$-colorable if its vertex set can be partitioned into $k$ parts so that the $i$th part induces a graph with maximum degree at most $d_i$ for all $i\in\{1, \ldots, k\}$. A class $\mathcal C$ of graphs is {\it balanced $k$-partitionable} and {\it unbalanced $k$-partitionable} if there exists a nonnegative integer $D$ such that all graphs in $\mathcal C$ are $(D, \ldots, D)$-colorable and $(0, \ldots, 0, D)$-colorable, respectively, where the tuple has length $k$. A set $X$ of cycles is a {\it cycle obstruction set} of a class $\mathcal C$ of planar graphs if every planar graph containing none of the cycles in $X$ as a subgraph belongs to $\mathcal C$. This paper characterizes all cycle obstruction sets of planar graphs to be balanced $k$-partitionable and unbalanced $k$-partitionable for all $k$; namely, we identify all inclusion-wise minimal cycle obstruction sets for all $k$.