Stability results on non spanning connected graphs

A graph G is t-spanning connected if between any two vertices of G there exist t(≥ 1) internally disjoint paths whose union is a spanning subgraph of G. This concept is a generalization of Hamiltonicity, as a graph is Hamiltonian(resp., Hamilton-connected) if and only if it is 2-spanning connected(resp., 1-spanning connected). Li and Ning [Spectral analougues of Erdős and Moon-Moser’s theorem-s on Hamilton cycles, Linear Multilinear A. (2016)], and independently, Füredi, Kostochka and Luo [A stability version for a theorem of Erdős on nonhamiltonian graphs, Discrete. Math. (2017)] proved a stability version of a theorem of Erdős on non 2-spanning connected graphs. In this paper, we generalize this result by providing stability versions for non t-spanning connected graphs for all t ≥ 1.