Characterization via independent efficient dominating sets

Characterize whether a graph G with finite ordinary chromatic number admits finite χ_{n,k}(G) for every integer k and positive integer n if and only if G admits an independent efficient dominating set.

Background

The paper proves that if a graph admits an independent efficient dominating set and has finite chromatic number, then χ{n,k}(G) exists for all integers k and positive integers n, with χ(G) ≤ χ{n,k}(G) ≤ χ(G)+1. The converse is left unresolved.

The proposed equivalence would connect the universal finiteness of closed chromatic numbers to the graph-theoretic structure of independent efficient domination. The authors note that finite chromatic number is necessary, since the infinite complete graph admits an independent efficient dominating set but has infinite ordinary chromatic number.

References

The following question asks if this is nearly equivalent to determining whether $G$ admits an IEDS. For a graph $G$, does $\chi_{n,k}(G) < \infty$ hold for all $k \in Z$ and $n\inZ+$ if and only if \mbox{$\chi(G)<\infty$} and $G$ admits an IEDS?

— Chromatic numbers with closed local modular constraints  (2503.00406 - Herden et al., 1 Mar 2025) in Question 2.7, immediately following Lemma 2.6, Section 2 (Basic Results)