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Keywords:
$\{K_{1,4},K_{1,4}+e\}$-free graph; neighborhood union; traceable
Summary:
A graph $G$ is called $\{H_1,H_2, \dots ,H_k\}$-free if $G$ contains no induced subgraph isomorphic to any graph $H_i$, $1\leq i\leq k$. We define $$\sigma _k= \min \biggl \{ \sum _{i=1}^k d(v_i) \colon \{v_1, \dots ,v_k\}\ \text {is an independent set of vertices in}\ G \biggr \}.$$ In this paper, we prove that (1) if $G$ is a connected $\{K_{1,4},K_{1,4}+e\}$-free graph of order $n$ and $\sigma _3(G)\geq n-1$, then $G$ is traceable, (2) if $G$ is a 2-connected $\{K_{1,4},K_{1,4}+e\}$-free graph of order $n$ and $|N(x_1)\cup N(x_2)|+|N(y_1)\cup N(y_2)|\geq n-1$ for any two distinct pairs of non-adjacent vertices $\{x_1,x_2\}$, $\{y_1,y_2\}$ of $G$, then $G$ is traceable, i.e., $G$ has a Hamilton path, where $K_{1,4}+e$ is a graph obtained by joining a pair of non-adjacent vertices in a $K_{1,4}$.
References:
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