The existence of a path-factor without small odd paths

In this paper, we show that if a graph $G$ satisfies $c_{1}(G-X)+\frac{2}{3}c_{3}(G-X)\leq \frac{4}{3}|X|+\frac{1}{3}$ for all $X\subseteq V(G)$, then $G$ has a $\{P_{2},P_{5}\}$-factor, where $c_{i}(G-X)$ is the number of components $C$ of $G-X$ with $|V(C)|=i$.


Introduction
In this paper, all graphs are finite and simple. Let G be a graph. We let V (G) and E(G) denote the vertex set and the edge set of G, respectively. For u ∈ V (G), we let N G (u) and d G (u) denote the neighborhood and the degree of u, respectively. For U ⊆ V (G), we let N G (U) = ( u∈U N G (u)) − U. For disjoint sets X, Y ⊆ V (G), we let E G (X, Y ) denote the set of edges of G joining a vertex in X and a vertex in Y .
For X ⊆ V (G), we let G[X] denote the subgraph of G induced by X. For two graphs H 1 and H 2 , we let H 1 + H 2 denote the join of H 1 and H 2 . Let P n denote the path of order n. For terms and symbols not defined here, we refer the reader to [2]. * e-mail:michitaka.furuya@gmail.com For a set H of connected graphs, a spanning subgraph F of a graph is called an H-factor if each component of F is isomorphic to a graph in H. A path-factor of a graph is a spanning subgraph whose components are paths of order at least 2. Since every path of order at least 2 can be partitioned into paths of orders 2 and 3, a graph has a path-factor if and only if it has a {P 2 , P 3 }-factor. Akiyama, Avis and Era [1] gave a necessary and sufficient condition for the existence of a path-factor (here i(G) denotes the number of isolated vertices of a graph G).
Theorem A (Akiyama, Avis and Era [1]) A graph G has a {P 2 , P 3 }-factor if and only if i(G − X) ≤ 2|X| for all X ⊆ V (G). Now we consider a path-factor with additional conditions. For example, one may require a path-factor to consist of components of large order. Concerning such a problem, Kaneko [3] gave a necessary and sufficient condition for the existence of a path-factor whose components have order at least 3. On the other hand, for k ≥ 4, it is not known that whether the existence problem of a path-factor whose components have order at least k is polynomially solvable or not, though some results about such a factor have been obtained (see, for example, Kano, Lee and Suzuki [4] and Kawarabayashi, Matsuda, Oda and Ota [5]).
In this paper, we study a different type of path-factor problem. Specifically, we focus on the existence of a {P 2 , P 2k+1 }-factor (k ≥ 2).
There are two motivations to study such factors. One of the motivations is related the notion of a hypomatchable graph. A graph H is hypomatchable if H − x has a perfect matching for every x ∈ V (H). A graph is a propeller if it is obtained from a hypomatchable graph H by adding new vertices a, b together with edge ab, and joining a to some vertices of H. Loebal and Poljak [6] proved the following theorem.
Theorem B (Loebal and Poljak [6]) Let H be a connected graph. If either H has a perfect matching, or H is hypomatchable, or H is a propeller, then the existence problem of a {P 2 , H}-factor is polynomially solvable. The problem is NP-complete for all other graphs H.
In particular, for k ≥ 2, the existence problem of a {P 2 , P 2k+1 }-factor is NPcomplete. Because of this fact, existence problems concerning {P 2 , P 2k+1 }-factors seem to have unjustly been ignored. However, in general, the fact that a problem is NP-complete in terms of algorithm does not mean that one cannot obtain a theoretical result concerning the problem. From this viewpoint, in this paper, we prove a theorem on the existence of a {P 2 , P 5 }-factor which, we hope, will serve as an initial attempt to develop the theory of {P 2 , P 2k+1 }-factors.
The other motivation is the fact that a {P 2 , P 2k+1 }-factor is a useful tool for finding large matchings. It is easy to see that if a graph G has a {P 2 , P 2k+1 }factor, then G has a matching M with |M| ≥ k 2k+1 |V (G)|. Thus the existence of a {P 2 , P 2k+1 }-factor helps to find large matchings.
In order to state our theorem, we need some more definitions. For a graph H, we let C(H) be the set of components of H, and for i ≥ 1, let Section 2). Thus if a condition concerning c 1 (G − X) and c 3 (G − X) for X ⊆ V (G) assures us the existence of a {P 2 , P 5 }-factor, then it will make a useful sufficient condition.
The main purpose of this paper is to prove the following theorem.
We prove Theorem 1.1 in Sections 3 and 4. In Subsection 5.1, we show that the bound 4 3 |X| + 1 3 in Theorem 1.1 is best possible. In our proof of Theorem 1.1, we make use of the following fact. We conclude this section with a conjecture concerning {P 2 , P 2k+1 }-factors with k ≥ 3. By Theorems A and 1.1, for k ∈ {1, 2}, there exists a constant a k > 1 such that the condition 0≤i≤k−1 c 2i+1 (G − X) ≤ a k |X| (X ⊆ V (G)) assures us the existence of a {P 2 , P 2k+1 }-factor (one can take a 1 = 2 and a 2 = 4 3 ). Thus one may expect that there exists a similar constant a k > 1 for k ≥ 3. However, when we consider the case where k ≥ 3 with k ≡ 0 (mod 3), the situation changes drastically; that is, there exist infinitely many graphs G having no {P 2 , P 2k+1 }-factor such that 0≤i≤k−1 c 2i+1 (G − X) ≤ 4k+6 8k+3 |X| + 2k+3 8k+3 for all X ⊆ V (G) (see Subsection 5.2). Thus we pose the following conjecture.

A necessary condition for a {P , P 5 }-factor
In this section, we give a necessary condition for the existence of a {P 2 , P 5 }-factor in terms of invariants c 1 and c 3 . We show the following proposition.
Proof. Let F be a {P 2 , P 5 }-factor of G, and let X ⊆ V (G). Then we can verify that Let C be a component of G − X with |V (C)| = 3 which does not belong to P ∈C(F ) C 3 (P − X). Then C intersects with at least two components of F − X. Since |V (C)| = 3, C contains a component of P − X of order 1 for some P ∈ C(F ). Since C is arbitrary, this implies that Thus we get the desired conclusion.

A path-factor in bipartite graph
Let G be a bipartite graph with bipartition (S, In this section, we focus on the existence of a special path-factor in bipartite graphs, and show the following theorem, which will be used in our proof of Theorem 1.1.
Theorem 3.1 Let S, T 1 and T 2 be disjoint sets with 1 ≤ |S| ≤ |T 1 | + |T 2 | and Let G be a bipartite graph with bipartition (S, T ) satisfying the property that for every X ⊆ V (G), we have either Before proving the theorem, we prove a lemma.

Lemma 3.2 Let S, T 1 , T 2 , T and G be as in Theorem 3.1. Then G has an S-central
G has a matching covering S by Hall's marriage theorem. In particular, G has an S-central subgraph F such that every component of F is a path of order at least 2.
Choose F so that |V (F )| is as large as possible.
Proof. Suppose that A contains a path which is not isomorphic to P 3 . Let i be the By the minimality of i, every path belonging to 1≤j≤i−1 A j is isomorphic to P 3 . Hence by the definition of A j , there exists a vertex v 3 backward and numbering the vertices accordingly) if necessary, we may assume that . Note that l ≥ 2 and l = 3. Thus by renumbering 1 } and every component of F ′ is a path of order at least 2, which contradicts the maximality of F .
We continue with the proof of the lemma. Let X 0 = ( A∈A V (A)) ∩ S and which contradicts the assumption that |T 1 | + 2 3 |T 2 | ≤ 4 3 |S| + 1 3 , completing the proof of the lemma.
We here outline the proof of Theorem 3.1. We choose an S-central path-factor F 0 so that F 0 will satisfy certain minimality conditions (see the paragraph following the proof of Claim 3.3). We then introduce operations which turn F 0 into a new path-factor (see the paragraphs following Claim 3.5 and Claim 3.6), and show that the new path-factor contradicts our choice of F 0 .
Proof of Theorem 3.1. We start with some definitions. Let F be an S-central path- If there is no fear of confusion, we simply write C i and C Figure 1). Figure 1: Edge ϕ F (AB) and vertices σ F (AB) and τ F (AB)

For a path
An admissible path P of D F is strongly admissible if P is not weakly admissible.
A path system with respect to F is a sequence (P 1 , . . . , P m ) (m ≥ 0) of admissible paths such that , P i is weakly admissible.
A path system (P 1 , . . . , P m ) with respect to F is complete if m ≥ 1 and P m is strongly admissible.
By straightforward calculations, we get the following claim (and we omit its proof).
The following claim plays a key role in the proof of the theorem.
3 (F ) = ∅, and let (P 1 , . . . , P m ) be a path system with respect to F (m ≥ 0). Then the system can be extend to a complete path system (P 1 , . . . , P m , P m+1 , . . . , P m ′ ) with respect to F . Proof. We take a maximal path system (P 1 , . . . , P m , P m+1 , . . . , P m ′ ) with respect to F . We show that (P 1 , . . . , P m ′ ) is a complete path system. Suppose that (P 1 , . . . , P m ′ ) is not a complete path system. Then P i is weakly admissible for each i with 1 ≤ i ≤ m ′ (this includes the case where m ′ = 0). and and and there exists an edge of D F from B i to an element in A 1 ∪ A 2 ∪ {B j | 1 ≤ j ≤ i − 1}. We choose (B 1 , . . . , B l ) so that l is as large as possible. Let
We turn to the proof of Theorem 3.1. By way of contradiction, suppose that C (1) 3 (F ) = ∅ for every S-central path-factor F of G. By Lemma 3.2, G has an Scentral path-factor F 0 . Note that an empty sequence is a path system with respect to F 0 . Hence by Claim 3.3, there exists a complete path system (P 1 , . . . , P m ) with respect to F 0 . Choose F 0 and (P 1 , . . . , P m ) so that As in the proof of Claim 3.1, by renumbering the vertices of some of the B i backward if necessary, we may assume that Note that (B3) means that when q 1 is even, the vertices of B 1 are numbered so that v 1,q 1 ∈ T . Thus v i,q i ∈ T for each i (1 ≤ i ≤ p). We can divide the type of B 1 into three possibilities as follows: Claim 3.4 One of the following holds: (1) |V (B 1 )| is even and s 1 is odd; Proof. If |V (B 1 )| is even, then (1) holds by (B3). Thus we may assume |V (B 1 )| is odd. Then by the definition of a strongly admissible path, As for B i with 2 ≤ i ≤ p − 1, the following claim follows immediately from the definition of a weakly admissible path. Claim 3.5 Let 2 ≤ i ≤ p − 1. Then one of the following holds: 3 (F 0 ) and s i = 2; (2) B i ∈ C 5 (F 0 ) and s i = 2 or 4; or Let i 0 be the minimum integer i (≥ 2) satisfying one of the following two conditions: (see Figure 2). Let 2 ≤ i ≤ i 0 − 1. By the definition of i 0 , t i ≥ 3. On the other hand, s i ≤ 4 by Claim 3.5. Hence (if B ′ 1 = ∅, then (3.11) trivially holds). Also because v i−1,s i−1 ∈ S and v i−1,q i−1 ∈ T . It follows from (3.12) that and Combining (3.10) through (3.15), we get the following claim.
This completes the proof of Theorem 3.1.

Proof of Theorem 1.1
we may assume that c 1 (G − X) + c 3 (G − X) = 0. By (4.1), empty or a path of order two. Therefore if we set By Claims 4.2(i) and 4.4, G has a {P 2 , P 5 }-factor.
This completes the proof of Theorem 1.1.

Examples
In this section, we construct graphs having no {P 2 , P 2k+1 }-factor.

Graphs without {P 2 , P 5 }-factor
Let n ≥ 1 be an integer. Let Q 0 be a path of order 3, and let a be an endvertex of Q 0 . Let Q 1 , . . . , Q n be disjoint paths of order 7, and for each i (1 ≤ i ≤ n), let b i be the center of Q i . Let H n denote the graph obtained from 0≤i≤n Q i by joining a to b i for every i (1 ≤ i ≤ n) (see Figure 4).
Suppose that H n has a {P 2 , P 5 }-factor F . Since Q 0 does not have a {P 2 , P 5 }factor, F contains ab i for some i (1 ≤ i ≤ n). Since d F (b i ) ≤ 2, this requires that at least one of the components of Q i − b i should have a {P 2 , P 5 }-factor, which is impossible because each component of Q i − b i is a path of order 3. Thus H n has no {P 2 , P 5 }-factor.
Proof. Let X ⊆ V (H n ). Then we can verify that and Since every component C of H n − X with |V (C)| = 1 belongs to 0≤i≤n C 1 (Q i − X), we have Let C be a component of H n − X with |V (C)| = 3 which does not belong to 0≤i≤n C 3 (Q i − X). Then C intersects with at least two of the Q i (0 ≤ i ≤ n). Since |V (C)| = 3, C contains a component of Q i − X of order 1 for some i (0 ≤ i ≤ n).
Since C is arbitrary, this implies that Thus we get the desired conclusion.
From Lemma 5.1, we get the following proposition, which implies that Theorem 1.1 is best possible.

Graphs without {P
Let k ≥ 3 be an integer with k ≡ 0 (mod 3), and write k = 3m. Let n ≥ 1 be an integer. Let R 0 be a complete graph of order n. For each i (1 ≤ i ≤ 2n + 1), let K i be a complete graph of order 2m − 1, and let R i denote the graph obtained from K i by joining each vertex of the union of 2m + 1 disjoint paths of order 2 to all vertices of K i . Let H ′ n = R 0 + ( 1≤i≤2n+1 R i ) (see Figure 5). Since |V (R i )| = 2k + 1 and R i does not contain a path of order 2k + 1, R i has no {P 2 , P 2k+1 }-factor. Suppose that H ′ n has a {P 2 , P 2k+1 }-factor F . Then for Figure 5: Graph H ′ n each i (1 ≤ i ≤ 2n + 1), F contains an edge joining V (R i ) and V (R 0 ). Since 2n + 1 > 2|V (R 0 )|, this implies that there exists x ∈ V (R 0 ) such that d F (x) ≥ 3, which is a contradiction. Thus H ′ n has no {P 2 , P 2k+1 }-factor.
From Lemma 5.3, we get the following proposition, which implies that if Conjecture 1 is true, then the coefficient of |X| in the conjecture is best possible.