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.
Eminjan Sabir (Tue,) studied this question.