An improvement of sufficient condition for k-leaf-connected graphs

For integer k≥2, a graph G is called k-leaf-connected if |V(G)|≥k+1 and given any subset S⊆V(G) with |S|=k,G always has a spanning tree T such that S is precisely the set of leaves of T. Thus a graph is 2-leaf-connected if and only if it is Hamilton-connected. In this paper, we present a best possible condition based upon the size to guarantee a graph to be k-leaf-connected, which not only improves the results of Gurgel and Wakabayashi (1986) and Ao et al. (2022), but also extends the result of Xu et al. (2021). Our key approach is showing that an (n+k−1)-closed non-k-leaf-connected graph must contain a large clique if its size is large enough. As applications, sufficient conditions for a graph to be k-leaf-connected in terms of the (signless Laplacian) spectral radius of G or its complement are also presented.