A note on Hamiltonicity of generalized net-free graphs of large diameter (Q1613470)

A finite simple graph \(G\) is said to be \(\text{CN}_{i,j,k}\)-free if no induced subgraph is isomorphic to the claw \(K_{1,3}\) or to the graph \(N_{i,j,k}\) obtained by identifying each vertex of a 3-circuit with an endvertex of one of three disjoint nontrivial paths of lengths \(i\), \(j\), and \(k\), respectively. Main result: let \(G\) be 2-connected and let \(j\geq 2\) be an integer. Then \(G\) is Hamiltonian if either (1) \(G\) is \(\text{CN}_{1,2,j}\)-free and diam\((G)\geq\max\{7,2j\}\) or (2) \(G\) is \(\text{CN}_{1,1,j}\)-free and diam\((G)\geq\max\{4,2j\}\).