The χ-stability index es χ (G) of a graph G is the minimum number of its edges whose removal results in a graph with the chromatic number smaller than that of G. In this paper three open problems from [European J. Combin. 84 (2020) 103042] are considered. Examples are constructed which demonstrate that a known characterization of k-regular (k ≤ 5) graphs G with es χ (G) = 1 does not extend to k ≥ 6. Graphs G with χ(G) = 3 for which es χ (G) + es χ (G) = 2 holds are characterized. Necessary conditions on graphs G which attain a known upper bound on es χ (G) in terms of the order and the chromatic number of G are derived. The conditions are proved to be sufficient when n ≡ 2 (mod 3) and χ(G) = 3.

The dichromatic number of D, denoted by − → χ (D), is the smallest integer k such that D admits an acyclic k-coloring. We use mader χ (F ) to denote the smallest integer kWe 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. B 153 (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.

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