Defective acyclic colorings of planar graphs

Abstract

AbstractThis paper studies two variants of defective acyclic coloring of planar graphs. For a graph and a coloring of , a 2‐colored cycle (2CC) transversal is a subset of that intersects every 2‐colored cycle. Let be a positive integer. We denote by the minimum integer such that has a proper ‐coloring which has a 2CC transversal of size , and by the minimum size of a subset of such that is acyclic ‐colorable. We prove that for any ‐vertex 3‐colorable planar graph and for any planar graph provided that . We show that these upper bounds are sharp: there are infinitely many planar graphs attaining these upper bounds. Moreover, the minimum 2CC transversal can be chosen in such a way that induces a forest. We also prove that for any planar graph and .