Degree sum conditions for traceable quasiclawfree graphs 
Received:March 07, 2021 Revised:June 09, 2021 
Key Word:
traceable graph quasiclawfree graphs degree sum

Fund ProjectL:National Natural Science Foundation of China (Grant No.\,11901268) 

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Abstract: 
A traceable graph is a graph containing a hamilton path. Let $N[v]=N(v)\cup\{v\}$ and $J(u,v)=\{w\in N(u)\cap N(v):N(w)\subseteq N[u]\cup N[v]\}$. A graph $G$ is called quasiclawfree if $J(u,v)\neq \emptyset$ for any $u,v\in V(G)$ with $d(u,v)=2$. Let $\sigma_{k}(G)=\min\{\sum_{v\in S}{\rm d}(v):S$ is an independent set of $V(G)$ with $S=k\}$.
In this paper, we prove that if $G$ is a connected quasiclawfree graph of order $n$ and $\sigma_{3}(G)\geq {n2}$, then $G$ is traceable; moreover, we give an example to show the bound in our result is best possible. We obtain that if $G$ is a connected quasiclawfree graph of order $n$ and $\sigma_{2}(G)\geq \frac{2({n2})}{3}$, then $G$ is traceable. 
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