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On hamiltonian properties of K1, r-free split graphs
Faculty Author(s): Zhan, Mingquan
Student Author(s):
Department: MATH
Publication: Discrete Mathematics
Year: 2023
Abstract: Let r ≥ 3 be an integer. A graph G is K 1 , r -free if G does not have an induced subgraph isomorphic to K 1 , 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 K 1 , 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 K 1 , r -free split graph is Hamilton-connected and every (r − 1) -connected K 1 , r -free split graph is fully cycle extendable.
Link: On hamiltonian properties of K1, r-free split graphs