A path factor of G is a spanning subgraph of G such that its each component is a path. A path factor is called a P≥n-factor if its each component admits at least n vertices. A graph G is called P≥n-factor covered if G admits a P≥n-factor containing e for any e ∈ E(G), which is defined by [Discrete Mathematics, 309, 2067–2076 (2009)]. We first define the concept of a (P≥n, k)-factor-critical covered graph, namely, a graph G is called (P≥n, k)-factor-critical covered if G-D is P≥n-factor covered for any D ⊆ V(G)with ∣D∣ = k. In this paper, we verify that (i) a graph G with k(G) ≥ k + 1 is (P⊆2, k)-factor-critical covered if bind $$\left( G \right) > {{2 + k} \over 3}$$ ; (ii) a graph G with ∣V(G)∣ ≥ k + 3 and k(G) ≥ k + 1 is (P≥3, k)-factor-critical covered if bind $$\left( G \right) \ge {{4 + k} \over 3}$$ .

中文翻译：

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图中具有给定性质的路径因子的一些存在定理

G 的路径因子是 G 的一个生成子图，使得它的每个分量都是一条路径。如果路径因子的每个分量至少允许 n 个顶点，则称为 P≥n 因子。图 G 被称为 P≥n-factor Covered 如果 G 承认包含 e 的 P≥n-factor 对于任何 e ∈ E(G)，由 [Discrete Mathematics, 309, 2067–2076 (2009)] 定义。我们首先定义一个(P≥n, k)-因子临界覆盖图的概念，即，如果GD是P≥n-因子覆盖，则图G被称为(P≥n, k)-因子-临界覆盖任意 D ⊆ V(G) 且 ∣D∣ = k。在本文中，我们验证 (i) k(G) ≥ k + 1 的图 G 是 (P⊆2, k)-因子临界覆盖如果 bind $$\left( G \right) > {{2 + k} \over 3}$$ ; (ii) ∣V(G)∣ ≥ k + 3 且 k(G) ≥ k + 1 的图 G 是 (P≥3, k)-因子临界覆盖，如果绑定 $$\left( G \right) \ge {{4 + k} \over 3}$$ 。