Signless Laplacian extremal graphs for color-k-critical graphs

Prove that for every integer k ≥ 2 and every color-k-critical graph F with chromatic number r+1 ≥ 4, every sufficiently large n-vertex F-free graph G satisfies q(G) ≤ q(H_{n,r,k}), with equality if and only if G is the graph H_{n,r,k}=K_{k-1} ∨ T_{n-k+1,r}.

Background

A color-k-critical graph is defined through an induced matching whose deletion lowers the chromatic number, together with a minimality condition involving deletion of vertices. This class properly contains color-critical graphs and includes examples such as the Petersen graph, Kneser graphs, and disjoint unions of cliques.

The paper recalls Simonovits’s edge-extremal theorem, according to which the unique sufficiently large extremal graph for a color-k-critical forbidden graph F is H_{n,r,k}=K_{k-1} ∨ T_{n-k+1,r}. It then proposes the signless Laplacian analogue for k ≥ 2. The authors explicitly explain that their proof of the main theorem cannot be extended because F-free graphs in this broader setting do not have the required degree-stable property, and no corresponding structural result is available.