I,F-partitions of Sparse Graphs

TL;DR

This paper proves that graphs with average degree less than 2.5 have a special vertex partition, leading to improved star 4-colorability results for sparse planar graphs with girth at least 10.

Contribution

It establishes a sharp bound on average degree for I,F-partitions in graphs, answering a key open question and improving star coloring results for sparse planar graphs.

Findings

Graphs with max average degree < 2.5 have I,F-partitions.

Planar graphs with girth ≥ 10 are star 4-colorable.

The result is sharp and improves previous bounds.

Abstract

A star $k$-coloring is a proper $k$-coloring where the union of two color classes induces a star forest. While every planar graph is 4-colorable, not every planar graph is star 4-colorable. One method to produce a star 4-coloring is to partition the vertex set into a 2-independent set and a forest; such a partition is called an I,F-partition. We use a combination of potential functions and discharging to prove that every graph with maximum average degree less than $\frac{5}{2}$ has an I,F-partition, which is sharp and answers a question of Cranston and West [A guide to the discharging method, arXiv:1306.4434]. This result implies that planar graphs of girth at least 10 are star 4-colorable, improving upon previous results of Bu, Cranston, Montassier, Raspaud, and Wang [Star coloring of sparse graphs, J. Graph Theory 62 (2009), 201-219].