Small feedback vertex sets in planar digraphs

Let $G$ be a directed planar graph on $n$ vertices, with no directed cycle of length less than $g\ge 4$. We prove that $G$ contains a set $X$ of vertices such that $G-X$ has no directed cycle, and $|X|\le \tfrac{5n-5}9$ if $g=4$, $|X|\le \tfrac{2n-5}4$ if $g=5$, and $|X|\le \tfrac{2n-6}{g}$ if $g\ge 6$. This improves recent results of Golowich and Rolnick.Comment: 5 pages, 1 figure - v3 final versio

Equitable partition of graphs into induced forests

An equitable partition of a graph $G$ is a partition of the vertex-set of $G$ such that the sizes of any two parts differ by at most one. We show that every graph with an acyclic coloring with at most $k$ colors can be equitably partitioned into $k-1$ induced forests. We also prove that for any integers $d\ge 1$ and $k\ge 3^{d-1}$, any $d$-degenerate graph can be equitably partitioned into $k$ induced forests. Each of these results implies the existence of a constant $c$ such that for any $k \ge c$, any planar graph has an equitable partition into $k$ induced forests. This was conjectured by Wu, Zhang, and Li in 2013.Comment: 4 pages, final versio

Precoloring co-Meyniel graphs

The pre-coloring extension problem consists, given a graph $G$ and a subset of nodes to which some colors are already assigned, in finding a coloring of $G$ with the minimum number of colors which respects the pre-coloring assignment. This can be reduced to the usual coloring problem on a certain contracted graph. We prove that pre-coloring extension is polynomial for complements of Meyniel graphs. We answer a question of Hujter and Tuza by showing that ``PrExt perfect'' graphs are exactly the co-Meyniel graphs, which also generalizes results of Hujter and Tuza and of Hertz. Moreover we show that, given a co-Meyniel graph, the corresponding contracted graph belongs to a restricted class of perfect graphs (``co-Artemis'' graphs, which are ``co-perfectly contractile'' graphs), whose perfectness is easier to establish than the strong perfect graph theorem. However, the polynomiality of our algorithm still depends on the ellipsoid method for coloring perfect graphs