Spanning Connectivity of K1,r-free Split Graphs

Let r >= 3 be an integer. A graph G is K-1,K-r-free if G does not have an induced subgraph isomorphic to K-1,K-r, and is a split graph if the vertex set of G can be partitioned into a clique and a stable set. It has been conjectured that for any r >= 3, every (r-1)-connected K-1,K-r-free split graph is hamiltonian and every r-connected K-1,K-r-free split graph is Hamilton-connected. These conjectures have been verified when r is an element of{3,4}. A graph G is spanning s-connected if for any pair of distinct vertices u,v is an element of V(G) and for all integers k with 0 <= k <= s, there exists a spanning subgraph of G consisting of k-internally disjoint (u,v)-paths. Thus G is spanning 1-connected implies that G is Hamilton-connected, and so is hamiltonian. We prove that for integers r and s with r is an element of{3,4} and s >= 3, each of the following holds for a K-1,K-r-free split graph G: (i) When r=3, G is s-connected if and only if G is spanning s-connected. (ii) When r=4, if G is 4-connected, then G is spanning 3-connected. (iii) If G is (r+2s-5)-connected, then G is spanning s-connected. These results (i) and (ii) extend former results on Hamilton-connectedness of 3-connected K-1,K-r-free split graphs in Liu et al. (2023). (c) 2024 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.