Defective acyclic colorings of planar graphs

TL;DR

This paper investigates defective acyclic colorings of planar graphs, establishing tight bounds on the size of edge sets intersecting cycles and making the graphs acyclically colorable with minimal edge removals.

Contribution

It introduces new bounds for the minimum size of edge sets intersecting cycles in defective acyclic colorings of planar graphs, with proofs of sharpness and structural properties.

Findings

For 3-colorable planar graphs, the minimum 2CC transversal size is at most n-3.

For all planar graphs, the 2CC transversal size for 4-colorings is at most n-5.

Bounds on minimal edge removals to achieve acyclic k-colorability are provided.

Abstract

This paper studies two variants of defective acyclic coloring of planar graphs. For a graph $G$ and a coloring $\varphi$ of $G$, a 2CC transversal is a subset $E'$ of $E(G)$ that intersects every 2-colored cycle. Let $k$ be a positive integer. We denote by $m_k(G)$ the minimum integer $m$ such that $G$ has a proper $k$-coloring which has a 2CC transerval of size $m$, and by $m'_k(G)$ the minimum size of a subset $E'$ of $E(G)$ such that $G-E'$ is acyclic $k$-colorable. We prove that for any $n$-vertex $3$-colorable planar graph $G$, $m_3(G) \le n - 3$ and for any planar graph $G$, $m_4(G) \le n - 5$ provided that $n \ge 5$. We show that these upper bounds are sharp: there are infinitely many planar graphs attaining these upper bounds. Moreover, the minimum 2CC transversal $E'$ can be chosen in such a way that $E'$ induces a forest. We also prove that for any planar graph $G$, $m'_3(G) \le (13n - 42) / 10$ and $m'_4(G) \le (3n - 12) / 5$.