Quadrangularly connected claw-free graphs
Let \(G_1\) be a graph arising from the cycle \(C_8\) with vertices \(x_1,x_2,\dots ,x_8\) by adding the edges of the cycle \(C_4\) with vertices \(x_1,x_3,x_5,x_7\) and let \(G_2\) be the graph obtained from \(G_1\) by adding the edge \(x_1x_5\). Let \(N(v,G)\) be the subgraph of a graph \(G\) induced by the set of all vertices adjacent to the vertex \(v\). \(G\) is quadrangularly connected if for every two edges \(e_1,e_2\) of \(G\) there is a sequence of cycles of length 3 or 4, \(C_1,C_2,\dots,C_r\), such that \(e_1\in E(C_1), e_2\in E(C_r)\) and \(E(C_i)\cap E(C_{i+1})\neq\emptyset\) for \(i=1,2,\dots,r-1\). It is shown that if \(G\) is a quadrangularly connected claw-free graph, which does not contain \(G_1\) or \(G_2\) as an induced subgraph and \(N(v,G)\) is disconnected for every vertex \(v\) of degree 4, then \(G\) is Hamiltonian.
scientific article; zbMATH DE number 5134189