Combinatorial characterization of ℓ-minimal graphs

Obtain a combinatorial characterization that determines exactly when a graph is [?] [?] [?] [?]

Background

An [?] [?] graph is a graph G satisfying r_+(G)=ℓ and |V(G)|=3ℓ. The paper proves that such graphs exist for every positive integer ℓ and constructs broad families of them, including stretched cliques in \hat{K}_{ℓ+2,ℓ-1} with clique number at most three. However, the paper also gives examples suggesting that membership in these stretched-clique families is not necessary. The unresolved question is therefore to identify a purely combinatorial criterion covering all ℓ-minimal graphs.

References

More ambitiously, can we obtain a combinatorial characterization of exactly when a given graph is $\ell$-minimal?

— Stable Set Polytopes with Rank $|V(G)|/3$ for the Lovász--Schrijver SDP Operator  (2501.07413 - Au et al., 13 Jan 2025) in Section 6, immediately following the first displayed Open Problem

However, it remains unclear for which graphs the agreement of the chromatic polynomial and the list-color function always persists.

\begin{question} Characterize the graphs $G$ such that for all integers $k$, $P(G,k)=P_{\ell}(G,k)>0$ implies $P(G,k+1)=P_{\ell}(G,k+1).$ \end{question}

— Non-persistence of equality between chromatic polynomials and list-color functions  (2608.19773 - Zhang et al., 20 Aug 2026) in Section 4, Concluding Remarks, immediately before the final Question environment

Is there such a condition, or is it too much to ask for?

— Matching algebras and their Dunkl subalgebras  (2609.34260 - Grinberg et al., 28 Sep 2026) in Section 5, subsection “Problems,” following the saturation implications