A path cover of a graph G is a spanning subgraph of G consisting of vertex disjoint paths; a minimum path cover of G is a path cover of G consisting of minimum number of paths; a path cover number of G, denoted by p(G), is the number of paths in a minimum path cover of G, i.e., \(p(G)=\min \{|{\mathcal {P}}|:\)\({\mathcal {P}}\) is a path cover of \(G\}.\) A graph G is quasi-claw-free if for any two vertices x, y with \(d(x,y)=2\) in G, there exists a vertex u with \(u\in N(x)\cap N(y)\) and \(N(u)\subseteq N[x]\cup N[y].\) In this paper, we prove that for any quasi-claw-free graph G of order n, if \(\sigma _{k+1}(G)\ge {n-k}\) for a positive integer k, then \(p(G)\le k,\) where \(\sigma _{k+1}(G)\) is the minimum degree sum of an independent set with \(k+1\) vertices in G. Our result is a generalization of Ore’s sufficient conditions of hamiltonicity for quasi-claw-free graphs.

中文翻译：

---

准无爪图中的最小路径覆盖

的曲线图的路径盖ģ是一个生成子图ģ由顶点不相交的路径; 的最小路径盖ģ是的路径盖ģ组成的路径最小数量; 的路径盖号码ģ，记为p（G ^），是的路径中的最小路径盖数量ģ，即\（P（G）= \分钟\ {| {\ mathcal {P}} |： \）\（{\ mathcal {P}} \）是的路径盖\（G \}。\）的曲线图G ^是准爪自由如果对任意两个顶点X，  ÿ与\（d（X， y）= 2 \）在G中，存在顶点ù与\（U \在N（x）的\帽N（Y）\）和\（N（U）\ subseteq N [X] \杯N [Y]。\）在本文中，我们证明对于阶数为n的任何准无爪图G，如果\（\ sigma _ {k + 1}（G）\ ge {nk} \）对于正整数k，则\（p（G） \文件K，\）其中\（\西格玛_ {K + 1}（G）\）是一个独立的组具有最小等级总和\（K + 1 \）在顶点ģ。我们的结果是对准无爪图的矿石的哈密尔顿性的充分条件的一般化。