Complete the conjecture on spectral conditions for [a,b]-factors

Prove the remaining cases of the conjecture on spectral conditions guaranteeing the existence of an [a,b]-factor in a graph, proposed by Cho, Hyun, O, and Park.

Background

The paper situates its spectral-radius results within prior work on factors. Fan, Lin, and Lu established that a graph G containing sufficiently many vertices and satisfying ρ(G) > ρ(H_{n,a}), where H_{n,a}=K_{a-1} ∨ (K_1 ∪ K_{n-a}), contains an [a,b]-factor. The authors explicitly describe this theorem as only a partial confirmation of an earlier conjecture concerning [a,b]-factors, so the full conjecture remains unresolved in the cited literature.

The paper’s main theorem addresses a rainbow k-factor problem and does not resolve the general [a,b]-factor conjecture itself. Accordingly, the unresolved task is to establish the conjecture in the cases not covered by the partial result of Fan, Lin, and Lu.