Some mader-perfect graph classes

Abstract

The dichromatic number of D, denoted by χ→(D), is the smallest integer k such that D admits an acyclic k-coloring. We use maderχ→(F) to denote the smallest integer k such that if χ→(D)≥k, then D contains a subdivision of F. A digraph F is called Mader-perfect if for every subdigraph F′ of F, maderχ→(F′)=|V(F′)|. We extend octi digraphs to a larger class of digraphs and prove that it is Mader-perfect, which generalizes a result of Gishboliner, Steiner and Szabó [Dichromatic number and forced subdivisions, J. Comb. Theory, Ser. B153 (2022) 1–30]. We also show that if K is a proper subdigraph of C↔4 except for the digraph obtained from C↔4 by deleting an arbitrary arc, then K is Mader-perfect.