Every 3-connected $\{K_{1,4},K_{1,4}+e\}$-free split graph of order at least 13 is Hamilton-connected

Abstract

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 $13$.