Some Mader-perfect graph classes

Abstract

The dichromatic number of $D$, denoted by $\overrightarrow{\chi }(D)$, is the smallest integer $k$ such that $D$ admits an acyclic $k$-coloring. We use $made{r}_{\overrightarrow{\chi }}(F)$ to denote the smallest integer $k$ such that if $\overrightarrow{\chi }(D)\ge k$, then $D$ contains a subdivision of $F$. A digraph $F$ is called Mader-perfect if for every subdigraph $F^{\prime }$ of $F$, ${\mathrm{mader}}_{\overrightarrow{\chi }}(F^{\prime })=|V(F^{\prime })|$. 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, {\it J. Comb. Theory, Ser. B} {\bf 153} (2022) 1--30]. We also show that if $K$ is a proper subdigraph of $\overleftrightarrow{C_4}$ except for the digraph obtained from $\overleftrightarrow{C_4}$ by deleting an arbitrary arc, then $K$ is Mader-perfect.

12 pages, 2 figures