Planar graphs without 4-cycles and close triangles are (2,0,0)-colorable

TL;DR

This paper proves that certain planar graphs, specifically those without 4-cycles and with limited triangle adjacency, can be colored with a partition where one part has maximum degree 2 and the others are independent sets.

Contribution

It establishes a new coloring result for a class of planar graphs with specific cycle and triangle adjacency restrictions.

Findings

Planar graphs without 4-cycles are (2,0,0)-colorable.

Graphs with no less than two edges between triangles are (2,0,0)-colorable.

The result extends understanding of graph colorability under structural constraints.

Abstract

For a set of nonnegative integers $c_1, \ldots, c_k$, a $(c_1, c_2,\ldots, c_k)$-coloring of a graph $G$ is a partition of $V(G)$ into $V_1, \ldots, V_k$ such that for every $i$, $1\le i\le k, G[V_i]$ has maximum degree at most $c_i$. We prove that all planar graphs without 4-cycles and no less than two edges between triangles are $(2,0,0)$-colorable.