Conditions for <i>k</i>-factor-critical graphs

For a nonnegative integer k , a graph G is said to be k -factor-critical if G − T has a perfect matching for any subset T ⊆ V ( G ) with |T | = k . In this paper, we first provide a condition in terms of the size of G to guarantee that G is k -factor-critical. For any graph G with minimum degree δ , we deduce a lower bound on the signless Laplacian spectral radius of G to ensure that G is k -factor-critical. Furthermore, we establish a condition based on Laplacian eigenvalues (resp. toughness) to determine whether a graph G is k -factor-critical. Additionally, we introduce a nullity condition for a t -connected graph to be k -factor-critical.