Two Forbidden Subgraph Pairs for Hamiltonicity of 3-Connected Graphs

For integers i, j, k with ${i\geq j\geq k\geq 0}$ , let N _{i, j, k} be the graph obtained by identifying end vertices of three disjoint paths of lengths i, j, k to the vertices of a triangle. In this paper, we show that every 3-connected {K _1,3, N _{i, 7-i, 2}}-free graph is hamiltonian, where ${i \in \{4,5\}}$ . This result is sharp in the sense that no one of the numbers i, 7?i and 2 in N _{i, 7-i, 2} can be replaced by a larger number.