In 1984, Matthews and Sumner conjectured that every 4-connected, claw-free graph contains a Hamiltonian cycle. This still unresolved conjecture has been the motivation for research into the existence of other cycle structures. In this paper, we consider the stronger property of pancyclicity for 4-connected graphs. In particular, we show that every 4-connected, $\{K_{1,3},N(i,j,k)\}$-free graph, where $i,j,k\ge 1$ and $i+j+k=6$, is pancyclic. This, together with results by Ferrara, Morris, Wenger, and Ferrara et al. completes a characterization of the graphs Y such that every $\{K_{1,3},Y\}$-free graph is pancyclic. In addition, this represents the best known progress towards answering a question of Gould concerning a characterization of the pairs of forbidden subgraphs that imply pancyclicity in 4-connected graphs.

1984 年，Matthews 和 Sumner 推测每个 4 连通的无爪图都包含一个哈密顿圈。这个仍未解决的猜想一直是研究其他循环结构存在的动机。在本文中，我们考虑了 4 连通图的泛循环性更强的特性。特别地，我们证明了每一个 4-connected，$\{{钾}_{1,3},N(\mathrm{一世},j,克)\}$- 自由图，其中 $\mathrm{一世},j,克\ge 1$ 和 $\mathrm{一世}+j+克=6$，是泛环的。这与费拉拉、莫里斯、温格和费拉拉等人的结果一起。完成图Y的表征，使得每个$\{{钾}_{1,3},是\}$-free 图是泛环的。此外，这代表了在回答 Gould 问题方面最著名的进展，该问题涉及在 4 连通图中暗示泛环性的禁用子图对的表征。