Planar Digraphs of Digirth Five Are 2‐Colorable

Abstract

Neumann-Lara (1985) andŠkrekovski conjectured that every planar digraph with digirth at least three is 2-colorable, meaning that the vertices can be 2-colored without creating any monochromatic directed cycles. We prove a relaxed version of this conjecture: every planar digraph of digirth at least five is 2-colorable. The result also holds in the setting of list colorings.

Small Feedback Vertex Sets in Planar Digraphs

“…Then G − X is acyclic. If g = 5 we have |X| |A| = |C| 2n−5 4 and if g 6, we have |X| |A| = |C| 2n−6 g , as desired. Assume now that g = 4.…”

“…A feedback vertex set in a digraph G is a set X of vertices such that G−X is acyclic, and the minimum size of such a set is denoted by τ (G). In this short note, we study the maximum f g (n) of τ (G) over all planar digraphs G on n vertices with digirth g. Harutyunyan [1,4] conjectured that f 3 (n) 2n 5 for all n. This conjecture was recently refuted by Knauer, Valicov and Wenger [5] who showed that f g (n) n−1 g−1 for all g 3 and infinitely many values of n. On the other hand, Golowich and Rolnick [3] recently proved that f 4 (n) 7n 12 , f 5 (n) 8n 15 , and f g (n) 3n−6 g for all g 6 and n. Harutyunyan and Mohar [4] proved that the vertex set of every planar digraph of digirth at least 5 can be partitioned into two acyclic subgraphs. This result was very recently extended to planar digraphs of digirth 4 by Li and Mohar [6], and therefore f 4 (n) n 2 .…”

“…Theorem 1. For all n 3 we have f 4 (n) 5n−5 9 , f 5 (n) 2n−5 4 and for all g 6, f g (n) 2n−6 g . In a planar graph, the degree of a face F , denoted by d(F ), is the sum of the lengths (number of edges) of the boundary walks of F .…”

List coloring digraphs

“…The list dichromatic number of D is the smallest integer k such that D is L ‐colorable for any k ‐list‐assignment L . We note that the definition of list coloring of digraphs is not quite new as it first appeared in where the authors derived an analog of Gallai's theorem for digraphs, as well as in . Our goal in this article is to initiate the study of the list dichromatic number.…”

Planar Digraphs without Large Acyclic Sets

“…An oriented graph is a digraph D without loops and multiple arcs. An acyclic set in D is a set of vertices that induces a directed subgraph without directed cycles.The complement Acyclic set 2n 5 [2] 5n + 6 12 [5] n 2 [4] n(g − 3) + 6 g [5] of an acyclic set of D is a feedback vertex set of D. A question of Albertson, which was the problem of the month on Mohar's web page [6] and was listed as a "Research Experience for Graduate Students" by West [11], asks whether every oriented planar graph on n vertices has an acyclic set of size at least n 2 . There are three independent strengthenings of this question in the literature.…”