Homotopy Covers of Graphs and Lifting Property

— The aim of this paper is creating requirements for a graph cover to have a Homotopy lifting property of topological space covers, or A-homotopy lifting property. Also, it defines the homotopy cover for a graph and gives a definition of the covering graph, and develops the path-lifting property.

as a space, these covering spaces a gain fail to recognize the structure of the graph, namely, that is vertices and edges. So there are covering graph, that is, graph homomorphism p: Ỹ → Y that maintain the local structure of the graphs. Specially, In particular, graph Ỹ should look like graph Y locally with the map formalizing the structure. In topology, given a covering space : Ũ → and continuous map : → also, there are also lifts ᾶ: → Ỹ , which factor through the space Ỹ there are lifting properties in topology that determine when a lift does or does not exist. Although the literature on A-Homotopy theory currently does not have an analogous name or set of attributes, the following three definitions provide a clearer understanding of what covering graphs are.

Definition 2.1[3]
A graph homomorphism ∝: 1 ( 2 ) that is adjacent vertices in 1 are mapped to the same vertex of 2 or adjacent vertices of 2 .

Definition 2.2
Let Y be a graph x ∈ V (Y). The closed neighborhood of x mean N[x] is the collection of vertices surrounding x as well as x itself exactly N[x] = {m ∈ V (Y)|{m, x} ∈ E (Y)or m = x}.

Definition 2.3[1]
The graph homomorphism : 1 → 2 is a local isomorphism if is onto and for each vertex x ∈ V ( 2 ) , also the vertex y ∈ −1 (x) iterative mapping P  We create a new subgraph with the restriction that it must be bijective on both its vertices and edges. For ∈ ( 1 ) let denoted the subgraph of 1 with a vertex on both vertices and edges. For z ∈ V( 1 ) let indicate the subgraph of 1 with vertex set V ( ) = N [z] and edge set E( ) = {{z, x} | x ∈ , x ≠ } .If : 1 → 2 is a local isomorphism then induces a graph homomorphism from the subgraph to the subgraph.
( ) For ∈ ( 1 ), In other words, a graph homomorphism exists. | : → ( ) This entails the following remark since it is bijective on the vertices and edges of the subgraphs.

3-Covering Graph Definition 3.1[1]
Let and Ỹ be graphed and let : Ỹ → be a graph homomorphism. If Is a local isomorphism the pair (Ỹ, ) is a covering graph of . Here, we illustrate some covering graph examples and explain how they vary from covering spaces.

Example 3.3
The graph homomorphism: P: 2 → is realized by P ([r]) = [r mod n] for all r ∈ [0, … , 2n − 1] then the pair ( 2 , p), forms a covering graph of the circle . The local isomorphism depicted in fig.1 is an example of covering graph of by 2 with k = 4 .Example 3.3 is like projecting the topological circle onto a different circle. Way that the first twice encircles the second. As we move further, we will define lift and discuss lifting qualities that are not covered in the literature that is currently available on A-Homotopy theory. The following definition is adapted from [9], except instead of a continuous map, it uses graph homomorphism.

Remark 4.2
Let ∞ denote the m-fold Cartesian product of ∞ we will only use nonbased graph homomorphisms ∝: ∞ → K with m = 1, 2. This will give us the path in graph and the graph Homotopy between the paths.

Definition 4.4[13]
If a graph homomorphism ∝: ∞ → stabilizes in every direction -r and + r for 1 ≤ r ≤ m then we say that α is a stable graph homomorphism .Let S m (Y) be the set of stable graph homomorphism .From the infinite m-cube ∞ to the graph .

Definition 4.5
Let be a graph and let (Ỹ, ) be the covering graph of Y given a graph homomorphism : G → Y a lift of is a graph homomorphism ᾶ: → Ỹ such that ∘ ᾶ = .