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

TL;DR

This paper improves the known size-based conditions for guaranteeing that a graph is $k$-leaf-connected, extending previous results and providing spectral radius criteria as applications.

Contribution

It presents a best possible size condition for $k$-leaf-connected graphs, improving prior bounds and extending spectral condition results.

Findings

Established a new size condition for $k$-leaf-connected graphs.

Extended spectral radius criteria for $k$-leaf-connectedness.

Improved upon previous results by Gurgel, Wakabayashi, Ao, Liu, Yuan, Li, Xu, Zhai, and Wang.

Abstract

For integer $k\geq2,$ a graph $G$ is called $k$-leaf-connected if $|V(G)|\geq k+1$ and given any subset $S\subseteq V(G)$ 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 [On $k$-leaf-connected graphs, J. Combin. Theory Ser. B 41 (1986) 1-16] and Ao, Liu, Yuan and Li [Improved sufficient conditions for $k$-leaf-connected graphs, Discrete Appl. Math. 314 (2022) 17-30], but also extends the result of Xu, Zhai and Wang [An improvement of spectral conditions for Hamilton-connected graphs, Linear Multilinear Algebra, 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.