Ryjáček et al.'s minimum-degree conjecture for forbidden-star graphs

Let $G$ be a graph. It is Hamiltonian if it contains a cycle through all its vertices. The graph $G$ is ${K_{1,4},K_{1,4}+e}$-free if it contains no induced subgraph isomorphic to either $K_{1,4}$ or the graph obtained from $K_{1,4}$ by adding an edge between two leaves; $\delta(G)$ denotes its minimum degree.

Ryjáček et al.'s conjecture. Every $4$-connected ${K_{1,4},K_{1,4}+e}$-free graph with minimum degree at least $6$ is Hamiltonian.

The conjecture extends the known result for $5$-connected graphs, where the minimum-degree bound $6$ suffices. Its status is not resolved in the supplied text.