Maximum average degree and relaxed coloring

TL;DR

This paper establishes a relationship between the maximum average degree of a graph and its relaxed coloring properties, providing conditions under which certain colorings are possible.

Contribution

It proves that graphs with maximum average degree less than a specific bound are colorable with a combination of degree-restricted and independent sets, extending coloring theory.

Findings

Graphs with mad(G) < 4/3 a + b are colorable with specified degree constraints.

The paper generalizes relaxed coloring conditions based on maximum average degree.

Provides a new bound linking maximum average degree to coloring feasibility.

Abstract

We say a graph is $(d, d, \ldots, d, 0, \ldots, 0)$-colorable with $a$ of $d$'s and $b$ of $0$'s if $V(G)$ may be partitioned into $b$ independent sets $O_1,O_2,\ldots,O_b$ and $a$ sets $D_1, D_2,\ldots, D_a$ whose induced graphs have maximum degree at most $d$. The maximum average degree, $mad(G)$, of a graph $G$ is the maximum average degree over all subgraphs of $G$. In this note, for nonnegative integers $a, b$, we show that if $mad(G)< \frac{4}{3}a + b$, then $G$ is $(1_1, 1_2, \ldots, 1_a, 0_1, \ldots, 0_b)$-colorable.