Stability results on non-spanning connected graphs

A graph [Formula: see text] is [Formula: see text]-spanning connected if between any two vertices of [Formula: see text] there exist [Formula: see text] internally disjoint paths whose union is a spanning subgraph of [Formula: see text]. This concept is a generalization of Hamiltonicity, as a graph is Hamiltonian (respectively, Hamilton-connected) if and only if it is [Formula: see text]-spanning connected (respectively, [Formula: see text]-spanning connected). Li and Ning [Spectral analougues of Erdős and Moon–Moser’s theorems on Hamilton cycles, Linear Multilinear Algebra 64 (2016) 1152–1169], and independently, Füredi et al. [A stability version for a theorem of Erdős on nonhamiltonian graphs, Discrete. Math. 340 (2017) 2688–2690] proved a stability version of a theorem of Erdős on non-[Formula: see text]-spanning connected graphs. In this paper, we generalize this result by providing stability versions for non-[Formula: see text]-spanning connected graphs for all [Formula: see text].