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. Ryjáček 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 Ryjáček et al.'s conjecture holds for split graphs of order at least $13$.

Figures (7)

• Figure 1: The graphs $K_{1,3},K_{1,4}~\text{and}~ K_{1,4}+e$.
• Figure 2: $V(G_1)\cap S=\{u,v\}$ and $V(G_2)\cap S=\{u,v,w\}$.
• Figure 3: For the proof Claim \ref{['cl1']}.
• Figure 4: $s=5$ and $t_i=2$ for $2\le i\le h$.
• Figure 5: The case when $d\in S_{2}\cup \{a_{1}^{3}\}$ in the proof of Claim \ref{['le51-cl3']}.

Theorems & Definitions (39)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Conjecture 1.8
• Conjecture 1.9
• Theorem 1.10