A graph $G$ is $\{ F_{ 1}, F_{ 2},\dots,F_{ k}\}$-free if $G$ contains no induced subgraph isomorphic to any $F_{ i}$ $(1\leq i \leq k)$. A connected graph $G$ is a split graph if its vertex set can be partitioned into a clique and an independent set. Ryj\'{ a}\v{ c}ek et al. [J. Comb. Theory, Ser. B 134 (2019) 239--263] conjectured that every $4$-connected $\{ K_{ 1,4},K_{ 1,4}+e\}$-free graph with minimum degree at least 6 is Hamiltonian and they confirmed the case with connectivity at least 5, where $K_{ 1,4}+e$ is the graph obtained from $K_{ 1,4}$ by adding a new edge. In this paper, we show that every 3-connected $\{ K_{ 1,4},K_{ 1,4}+e\}$-free split graph of order at least $13$ is Hamilton-connected. It implies that Ryj\'{ a}\v{ c}ek et al.'s conjecture holds for split graphs of order at least $13$.
Categories: math.CO