Fractional vertex-arboricity of planar graphs

TL;DR

This paper explores the fractional vertex-arboricity of planar graphs, connecting it to longstanding conjectures about the size of induced forests, and proves a new upper bound for graphs with girth at least five.

Contribution

It introduces the fractional vertex-arboricity concept for planar graphs and provides progress on a conjecture related to graphs of girth at least five.

Findings

Proves that planar graphs of girth at least five have fractional vertex-arboricity at most 2 - 1/324.

Connects fractional vertex-arboricity to classical conjectures on induced forests.

Advances understanding of induced forest sizes in planar graphs.

Abstract

We initiate a systematic study of the fractional vertex-arboricity of planar graphs and demonstrate connections to open problems concerning both fractional coloring and the size of the largest induced forest in planar graphs. In particular, the following three long-standing conjectures concern the size of a largest induced forest in a planar graph, and we conjecture that each of these can be generalized to the setting of fractional vertex-arboricity. In 1979, Albertson and Berman conjectured that every planar graph has an induced forest on at least half of its vertices, in 1987, Akiyama and Watanabe conjectured that every bipartite planar graph has an induced forest on at least five-eighths of its vertices, and in 2010, Kowalik, Lu\v{z}ar, and \v{S}krekovski conjectured that every planar graph of girth at least five has an induced forest on at least seven-tenths of its vertices. We make progress toward the fractional generalization of the latter of these, by proving that every planar graph of girth at least five has fractional vertex-arboricity at most $2 - 1/324$.