Some existence theorems on path-factor critical avoidable graphs

A spanning subgraph $F$ of $G$ is called a path factor if every component of $F$ is a path of order at least 2. Let $k\geq2$ be an integer. A $P_{\geq k}$-factor of $G$ means a path factor in which every component has at least $k$ vertices. A graph $G$ is called a $P_{\geq k}$-factor avoidable graph if for any $e\in E(G)$, $G$ has a $P_{\geq k}$-factor avoiding $e$. A graph $G$ is called a $(P_{\geq k},n)$-factor critical avoidable graph if for any $W\subseteq V(G)$ with $|W|=n$, $G-W$ is a $P_{\geq k}$-factor avoidable graph. In other words, $G$ is $(P_{\geq k},n)$-factor critical avoidable if for any $W\subseteq V(G)$ with $|W|=n$ and any $e\in E(G-W)$, $G-W-e$ admits a $P_{\geq k}$-factor. In this article, we verify that (\romannumeral1) an $(n+r+2)$-connected graph $G$ is $(P_{\geq2},n)$-factor critical avoidable if $I(G)>\frac{n+r+3}{2(r+2)}$; (\romannumeral2) an $(n+r+2)$-connected graph $G$ is $(P_{\geq3},n)$-factor critical avoidable if $t(G)>\frac{n+r+2}{2(r+2)}$; (\romannumeral3) an $(n+r+2)$-connected graph $G$ is $(P_{\geq3},n)$-factor critical avoidable if $I(G)>\frac{n+3(r+2)}{2(r+2)}$; where $n$ and $r$ are two nonnegative integers.


Introduction
In this work, we discuss only finite, undirected and simple graphs.We denote by G = (V (G), E(G)) a graph, where V (G) denotes the vertex set of G and E(G) denotes the edge set of G.For a vertex x of G, the degree of x in G, denoted by d G (x), is the number of vertices adjacent to x in G.For a vertex subset X of G, G[X] denotes the subgraph of G induced by X, and G − X denotes the subgraph derived from G by removing all vertices in X.For an edge subset E of G, G − E denotes the subgraph acquired from G by deleting all edges in E .For a vertex (or an edge) subset Q, we denote G − Q by G − u for convenience if Q = {u}.Let i(G), ω(G) and κ(G) denote the number of isolated vertices, the number of connected components and the vertex connectivity of G, respectively.We use K n and P n to denote the complete graph and the path with n vertices, respectively.Let G 1 and G 2 be two graphs.Then the join G 1 + G 2 denotes the graph with vertex set V (G 1 + G 2 ) = V (G 1 ) ∪ V (G 2 ) and edge set 1 arXiv:2304.00937v1[math.CO] 3 Apr 2023 The toughness of a graph G, denoted by t(G), was first introduced by Chvátal [3].If G is not complete, then otherwise, t(G) = +∞.
The isolated toughness of a graph G, denoted by I(G), was first introduced by Yang, Ma and Liu [19].If G is not complete, then otherwise, I(G) = +∞.A spanning subgraph F of G is called a path factor if every component of F is a path of order at least 2. Let k ≥ 2 be an integer.A P ≥k -factor of G means a path factor in which every component has at least k vertices.
Las Vergnas [14] showed a necessary and sufficient condition for graphs to possess P ≥2 -factors.

Theorem 1 ( [14]
).A graph G possesses a P ≥2 -factor if and only if G satisfies R is the corona of a factor-critical graph H with at least three vertices, namely, R is acquired from H by adding a new vertex z = z(y) together with a new edge yz for any y ∈ V (H) to H (Figure 1, which was shown by Kano, Lu and Yu [12]).We easily see that d R (z) = 1.In particular, a sun with at least six vertices is called a big sun.Let sun(G) denote the number of sun components of G.In fact, i(G) ≤ sun(G) ≤ ω(G).
Kaneko [9] gave a necessary and sufficient condition for graphs admitting P ≥3 -factors.Kano, Katona and Király [10] gave a simple proof.
Theorem 2 ( [9,10]).A graph G contains a P ≥3 -factor if and only if G satisfies sun(G − X) ≤ 2|X| for every vertex subset X of G.
In recent years, many results on path factors were derived.Kelmans [13] raised some results on the existence of path factors in claw-free graphs.Ando et al [1] derived a minimum degree condition for a claw-free graph to have a path factor.Kano, Lee and Suzuki [11] verified that every connected cubic bipartite graph with at least eight vertices admits a P ≥8 -factor.Egawa and Furuya [4] showed some sufficient conditions for graphs to have path factors.Kano, Lu and Yu [12] presented a sufficient condition for the existence of P ≥3 -factor.Wu [15], Zhou et al [25,28,[31][32][33] derived some sufficient conditions for graphs to possess P ≥3 -factors with given properties.Gao, Wang and Chen [8] posed some tight bounds for the existence of P ≥3 -factors in graphs.Dauer, Katona, Kratsch and Veldman [2], Gao, Guirao and Chen [5], Liu and Zhang [16] established some relationships between toughness and graph factors.Gao and Wang [7], Gao, Liang and Chen [6] established some relationships between isolated toughness and graph factors.More results on graph factors were acquired by Zhou [21][22][23][24]26], Zhou and Liu [29], Zhou, Xu and Sun [30], Wang and Zhang [17,18], Yuan and Hao [20].
A Zhou [27] acquired some toughness or isolated toughness conditions for graphs to be (P ≥k , n)-factor critical avoidable graphs for k = 2, 3.

(P ≥, n)-factor critical avoidable graphs
In this section, we pose a sufficient conditions using isolated toughness for graphs to be (P ≥2 , n)-factor critical avoidable graphs, which is an improvements of Theorem 3 for n ≥ 1. Theorem 6.Let n and r be two nonnegative integers, and let G be an (n + r + 2)-connected graph.If its isolated toughness I(G) > n+r+3 2(r+2) , then G is (P ≥2 , n)-factor critical avoidable.Proof.Obviously, Theorem 6 holds for a complete graph.Next, we assume that G is not a complete graph.Let H = G − W − e for any W ⊆ V (G) with |W | = n and any e ∈ E(G − W ). It suffices to claim that H has a P ≥2 -factor.Suppose that H has no P ≥2 -factor.Then it follows from Theorem 1 that for some The following proof is divided into three cases.
In this case, it is obvious that there exists a Then by ( 1) and Claim 1, we deduce and so In this case, there exists a vertex u such that d G−W −X (u) = 1.Let v be an unique vertex adjacent to u in G − W − X, and e = uv.Similar to this discussion of Case 1, we easily deduce and so According to (1), we obtain If n = 0, then from (2) we have then it follows from (2) and Claim 1 that . This completes the proof of Theorem 6. Remark 1. Next, we show that the condition on I(G) in Theorem 6 is sharp. Let , where n and r are two nonnegative integers with n ≥ r + 1.Clearly, G is (n + r + 2)-connected and In terms of Theorem 1, G − W − e has no P ≥2 -factor.Hence, G is not (P ≥2 , n)-factor critical avoidable.
Remark 2. Next, we explain that the condition on (n + r + 2)-connected in Theorem 6 is best possible. Let , where n and r are two nonnegative integers with n ≥ r.It is obvious that G is (n + r + 1)-connected and In terms of Theorem 1, G − W − e has no P ≥2 -factor.So G is not (P ≥2 , n)-factor critical avoidable.

(P ≥, n)-factor critical avoidable graphs
We first verify the following lemma.
Lemma 1.Let n and r be two nonnegative integers, let G be an (n + r + 2)-connected graph, and let On the other hand, since G is (n + r + 2)-connected, H is (r + 1)-connected.Thus, we have sun(H) ≤ Hence, we obtain sun(H) = 1.Combining this with H being (r + 1)-connected, H is a sun.Note that G is (n + r + 2)-connected, and so which implies that H is a big sun.Hence, there exist at least three vertices with degree 1 in H, and so there exists at least one vertex v with d G−W (v) = 1.Thus, we acquire In what follows, we consider 1 ≤ |X| ≤ r + 1.
Next, we raise two sufficient conditions using toughness and isolated toughness for graphs being (P ≥3 , n)-factor critical avoidable graphs, which are the improvements of Theorems 4 and 5.
Theorem 7. Let n and r be two nonnegative integers, and let G be an (n + r + 2)-connected graph.If its toughness t(G) > n+r+2 2(r+2) , then G is (P ≥3 , n)-factor critical avoidable.Proof.For a complete graph G, Theorem 7 is true.In the following, we assume that G is not a complete graph.Let H = G − W − e for any W ⊆ V (G) with |W | = n and any e ∈ E(G − W ). It suffices to prove that H admits a P ≥3 -factor.On the contrary, we assume that H has no P ≥3 -factor.In view of Theorem 2, we obtain for some subset X of V (H).
In view of (1) and Lemma 1, we infer Combining this with the definition of t(G), we get which contradicts t(G) > n+r+2 2(r+2) .We finish the proof of Theorem 7. Remark 3. Next, we show that the condition on t(G) in Theorem 7 is sharp. Let , where n and r are two nonnegative integers.Obviously, G is In terms of Theorem 2, G − W − e has no P ≥3 -factor.Hence, G is not (P ≥3 , n)-factor critical avoidable.Remark 4. Next, we explain that the condition on (n + r + 2)-connected in Theorem 7 cannot be replaced by (n + r + 1)-connected. Let , where n ≥ 1 and r ≥ 0 are two integers.We see that G is (n+r+1)-connected and t(G) In terms of Theorem 2, G−W −e has no P ≥3 -factor.Therefore, G is not (P ≥3 , n)-factor critical avoidable.Theorem 8. Let n and r be two nonnegative integers, and let G be an (n + r + 2)-connected graph.If its isolated toughness I(G) > n+3(r+2) 2(r+2) , then G is (P ≥3 , n)-factor critical avoidable.Proof.It is obvious that Theorem 8 is true for a complete graph.In what follows, we assume that G is not complete.Let H = G − W − e for any W ⊆ V (G) with |W | = n and any e = uv ∈ E(G − W ). It suffices to verify that H contains a P ≥3 -factor.By means of contrary, we assume that H has no P ≥3 -factor.Then by Theorem 2, we admit for some vertex subset X of H.
Suppose that there exist a isolated vertices, b K 2 's and c big sun components where Let R i be the factor-critical subgraph of H i .We select one vertex from every K 2 component of H − X, and denote the set of such vertices by Y .Thus, we admit According to ( 1), ( 2), Lemma 1 and In The following proof is divided into four cases.
In this case, u ∈ V (aK 1 ) and v ∈ V (bK 2 ), or u ∈ V (aK 1 ) and v ∈ V (H i ), or u, v belong to two different K 2 components, or u ∈ V (H i ) and v ∈ V (bK 2 ), or v ∈ V (H i ) and u ∈ V (H j ) (i = j).Proof.We consider the following two subcases.
In terms of (3) and the definition of I(G), we get Using (3) and the definition of I(G), we derive This completes the proof of Claim 1.
In light of (3), Claim 1 and Lemma 1, we obtain In this case, u, v ∈ V (aK 1 ), or u ∈ V (M ) and v ∈ V (Q), where M is a sun component of H − X, Q is a non-sun component of H − X, and M ∪ Q ∪ {e} is a non-sun component of (H − X) ∪ {e}.Subcase 2.1.u, v ∈ V (aK 1 ).

Figure 1 :
Figure 1: A factor-critical graph H and the sun R obtained from H.
graph G is called a P ≥k -factor avoidable graph if for any e ∈ E(G), G has a P ≥k -factor avoiding e.A graph G is called a (P ≥k , n)-factor critical avoidable graph if for any W ⊆ V (G) with |W | = n, G − W is a P ≥k -factor avoidable graph.In other words, G is (P ≥k , n)-factor critical avoidable if for any W ⊆ V (G) with |W | = n and any e ∈ E(G − W ), G − W − e contains a P ≥k -factor.