Characterization for higher chromatic number

Characterize, for each integer k≥4, the connected graphs G with chromatic number k that do not realize the sequence (2,1^∞).

Background

The sequence (2,1∞) requires one color class to contain at least two color-dominating vertices while every other used color class contains at least one. The paper completely characterizes the connected graphs with chromatic number 3 that fail to realize this sequence: they are C5 and the graphs T_{r,s,t}.

The corresponding classification for graphs of chromatic number at least 4 is explicitly stated to be unknown, and the question asks for a characterization separately for every k≥4.

References

For each integer $k\geq4$, which connected graphs $G$ with $\chi(G)=k$ do not realize the sequence $(2,1\infty)$?

— Sequence b-colorings in graphs  (2609.08484 - Jakovac et al., 8 Sep 2026) in Question 3, Section 7 (Concluding remarks and open questions)