Vertex partitions of $(C_3,C_4,C_6)$-free planar graphs

TL;DR

This paper proves that all $(C_3,C_4,C_6)$-free planar graphs can be partitioned into a 0-degree and a 6-degree subgraph, but determining if they can be partitioned into a 0-degree and a 3-degree subgraph is NP-complete.

Contribution

It establishes a universal $(0,6)$-colorability for $(C_3,C_4,C_6)$-free planar graphs and proves NP-completeness for $(0,3)$-colorability decision.

Findings

Every $(C_3,C_4,C_6)$-free planar graph is $(0,6)$-colorable.

Deciding $(0,3)$-colorability for these graphs is NP-complete.

Abstract

A graph is $(k_1,k_2)$-colorable if its vertex set can be partitioned into a graph with maximum degree at most $k_1$ and and a graph with maximum degree at most $k_2$. We show that every $(C_3,C_4,C_6)$-free planar graph is $(0,6)$-colorable. We also show that deciding whether a $(C_3,C_4,C_6)$-free planar graph is $(0,3)$-colorable is NP-complete.