Planar graphs with distance of 3-cycles at least 2 and no cycles of lengths 5, 6, 7

TL;DR

This paper proves that certain planar graphs with specific cycle restrictions are weakly 2-degenerate and can be partitioned into an independent set and a forest, advancing understanding of their coloring properties.

Contribution

It establishes that planar graphs with distance 3-cycle at least 2 and no cycles of lengths 5, 6, 7 are weakly 2-degenerate and admits a specific vertex partition.

Findings

Such graphs are weakly 2-degenerate.

They can be partitioned into an independent set and a forest.

This contributes to graph coloring theory.

Abstract

Weak degeneracy of a graph is a variation of degeneracy that has a close relationship to many graph coloring parameters. In this article, we prove that planar graphs with distance of $3$-cycles at least 2 and no cycles of lengths $5, 6, 7$ are weakly $2$-degenerate. Furthermore, such graphs can be vertex-partitioned into two subgraphs, one of which has no edges, and the other is a forest.