On hamiltonian properties of K1,r-free split graphs

Abstract

Let r≥3 be an integer. A graph G is K1,r-free if G does not have an induced subgraph isomorphic to K1,r. A graph G is fully cycle extendable if every vertex in G lies on a cycle of length 3 and every non-hamiltonian cycle in G is extendable. A connected graph G is a split graph if the vertex set of G can be partitioned into a clique and a stable set. Dai et al. (2022) [4] conjectured that every (r−1)-connected K1,r-free split graph is hamiltonian, and they proved this conjecture when r=4 while Renjith and Sadagopan proved the case when r=3. In this paper, we introduce a special type of alternating paths in the study of hamiltonian properties of split graphs and prove that a split graph G is hamiltonian if and only if G is fully cycle extendable. Consequently, for r∈{3,4}, every r-connected K1,r-free split graph is Hamilton-connected and every (r−1)-connected K1,r-free split graph is fully cycle extendable.