Institute of Mathematics

Abstract:  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 $\sigma_k= \min \{ \sum_{i=1}^k d(v_i) \colon\{v_1, \dots,v_k\}$ is an independent set of vertices in $G \}$. 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 $K_{1,4}$.