On k-leaf-connected graphs

AbstractA graph G is Hamilton-connected if given any two vertices u and v of G, there is a Hamilton path in G with ends u and v. In this note we consider a generalization of this property. For k ≥ 2 we say that a graph G = (VG, EG) is k-leaf-connected if |VG| > k and given any subset S of VG with |S| = k, G has a spanning tree T such that the set S is the set of endvertices of T. Thus a graph is 2-leaf-connected if and only if it is Hamilton-connected. This generalization is due to U. S. R. Murty. We prove that the k-leaf-connectedness property is (|VG| + k − 1)-stable, give sufficient conditions for a graph to be k-leaf-connected, present some necessary conditions and other related results. We show that for all naturals n, k, 2 ≤ k < n − 2, there is a sparse k-leaf-connected graph of order n

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

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.Comment: 15 pages, 2 figure

Further topics in connectivity

Continuing the study of connectivity, initiated in §4.1 of the Handbook, we survey here some (sufficient) conditions under which a graph or digraph has a given connectivity or edge-connectivity. First, we describe results concerning maximal (vertex- or edge-) connectivity. Next, we deal with conditions for having (usually lower) bounds for the connectivity parameters. Finally, some other general connectivity measures, such as one instance of the so-called “conditional connectivity,” are considered. For unexplained terminology concerning connectivity, see §4.1.Peer ReviewedPostprint (published version