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

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.

14 pages, 5 figures

References in corpus (2)

• Weak degeneracy of planar graphs without 4- and 6-cycles
• Toroidal graphs without $K_5^{-}$ and 6-cycles