SUFFICIENT CONDITIONS FOR GRAPHS WITH { 𝑃 2 , 𝑃 5 } -FACTORS

. For a graph 𝐺 , a spanning subgraph 𝐹 of 𝐺 is called an { 𝑃 2 , 𝑃 5 } -factor if every component of 𝐹 is isomorphic to 𝑃 2 or 𝑃 5 , where 𝑃 𝑘 denotes the path of order 𝑘 . It was proved by Egawa and Furuya that if 𝐺 satisfies 3 𝑐 1 ( 𝐺 − 𝑆 ) + 2 𝑐 3 ( 𝐺 − 𝑆 ) ≤ 4 | 𝑆 | + 1 for all 𝑆 ⊆ 𝑉 ( 𝐺 ), then 𝐺 has a { 𝑃 2 , 𝑃 5 } - factor, where 𝑐 𝑘 ( 𝐺 − 𝑆 ) denotes the number of components of 𝐺 − 𝑆 with order 𝑘 . By this result, we give some other sufficient conditions for a graph to have a { 𝑃 2 , 𝑃 5 } -factor by various graphic parameters such as toughness, binding number, degree sums, etc. Moreover, we obtain some regular graphs and some 𝐾 1 ,𝑟 -free graphs having { 𝑃 2 , 𝑃 5 } -factors.


Introduction
In this paper, we consider only finite and undirected graph without loops or multiple edges.Other basic graph-theoretic terminologies not defined here can be found in [4].Let  = ( (), ()) be a graph, where  () and () denote the vertex set and the edge set of , respectively.A spanning subgraph of  is a subgraph  of  such that  () =  () and () ⊆ ().For  ⊆  (),  −  denotes the graph obtained from  by deleting all the vertices of  and [] denotes the subgraph of  induced by .For  ∈  (), we use   () and   () to denote the degree of  and the set of vertices adjacent to  in , respectively.For  ⊆  (), we write   () = ∪ ∈   ().A graph  is said to be -regular if every vertex of  has degree .We denote the minimum degree and the number of connected components of a graph  by () and (), respectively.Define  2 () = min{  () +   () : {, } ⊆  () is an independent set of }.
For a connected graph , its toughness, denoted by  (), was first introduced by Chvátal [5] as follows.If  is complete, then  () = +∞; otherwise, The binding number is introduced by Woodall [21] and defined as }︂ .
The complete bipartite graph  1, is called the star of order  +1.We call a graph  is  1, -free if  does not contain an induced subgraph isomorphic to  1, .In particular, a graph is said to be claw-free if it is  1,3 -free.
For a family of connected graphs ℱ, a spanning subgraph  of a graph  is called an ℱ-factor of  if each component of  is isomorphic to some graph in ℱ.Let   denote the path of order .A spanning subgraph of a graph  is called a { 2 ,  5 }-factor of  if its each component is isomorphic to  2 or  5 .Similarly, { 2 ,  3 }-factor means a graph factor in which every component is a path of order exactly two or three.
Akiyama et al. [2] demonstrated the following classical result, which is a criterion for graphs with { 2 ,  3 }factors.We denote by () the number of isolated vertices of a graph .
For an integer  ≥ 2, a {  :  ≥ }-factor is briefly denoted by  ≥ -factor.Note that a graph has  ≥2 -factors if and only if it has { 2 ,  3 }-factors.Kaneko [13] gave a necessary and sufficient condition for the existence of  ≥3 -factors.For  ≥ 4, it is not known that whether the existence problem of  ≥ -factors is polynomially solvable or not, though some results about such factors on special classes of graphs have been obtained (see, for example, Kano et al. [16], Ando et al. [3], and Kawarabayashi et al. [17]).
A graph  is hypomatchable if  −  has a perfect matching for every  ∈  ().A graph is a propeller if it is obtained from a hypomatchable graph  by adding new vertices ,  and edge , and joining  to some vertices of .Loebal and Poljak [18] proved the following theorem.

Theorem 1.2 ([18]
).Let  be a connected graph.If  has a perfect matching,  is hypomatchable, or  is a propeller, then the existence problem of a { 2 , }-factor is polynomially solvable.The problem is NP-complete for all other graphs .
For  ⊆  (), let   ( − ) be the set of components of order  in  − , where integer
Although a sufficient condition for the existence of { 2 ,  5 }-factors was proposed by Egawa and Furuya, to check the condition in Theorem 1.4 is a non-trivial task.This paper is attempted to find more sufficient conditions for the existence of { 2 ,  5 }-factors using various graphic parameters, or to determine special classes of graphs to have { 2 ,  5 }-factors such as -regular graphs, planar graphs and  1, -free graphs.The graphic parameters been studied in this paper include minimum degree, toughness, binding number, etc. 2. Proof of Theorem 1.5 Suppose, to the contrary, that  is a connected graph of order  ≥ 4 and contains no { 2 ,  5 }-factor.By Theorem 1.4, there exists  ⊆  () such that 3 1 ( − ) + 2 3 ( − ) > 4|| + 1. Due to the integrality, we obtain Proof.Suppose that  = ∅, then by (2.1), we have Obviously,  has a { 2 ,  5 }-factor which is also a { 2 ,  5 }-factor of , a contradiction.In the following, we assume that  is not complete.By Claim 2.1 and (2.1), we have that By the definition of  (), it follows that This contradiction completes the proof of Statement (i) of Theorem 1.5.(ii) We choose one vertex from each component of  −  with order 3, and denote by  ′ the set of such vertices.