Defective 2-colorings of planar graphs without 4-cycles and 5-cycles

TL;DR

This paper investigates the complexity and colorability of planar graphs without 4- and 5-cycles, proving NP-completeness for certain coloring problems and establishing specific colorability results.

Contribution

It demonstrates NP-completeness of (0,k)-colorability and constructs non-$(1,k)$-colorable graphs, advancing understanding of defective colorings in restricted planar graphs.

Findings

NP-complete for (0,k)-colorability in such graphs

Existence of non-$(1,k)$-colorable planar graphs without 4- and 5-cycles

Certain $(d_1,d_2)$-colorability results proven

Abstract

Let $G$ be a graph without 4-cycles and 5-cycles. We show that the problem to determine whether $G$ is $(0,k)$-colorable is NP-complete for each positive integer $k.$ Moreover, we construct non-$(1,k)$-colorable planar graphs without 4-cycles and 5-cycles for each positive integer $k.$ Finally, we prove that $G$ is $(d_1,d_2)$-colorable where $(d_1,d_2)=(4,4), (3,5),$ and $(2,9).$