The homotopy exact sequence of a pair of graphs

Higher homotopy of graphs has been defined in several articles. However, the existence of a long exact sequence associated to a pair (G, A) has not been touched at. We treat it here. Applied to the discrete spheres, this lead to interesting open questions.


Introduction
Homotopy groups of graphs have been defined in Benayat and Kadri (1997) and Babson et al. (2006).One of the main property of any consistent homotopy theory is the existence of a long exact sequence associated to any based pair   0 , , G H v of graphs.In this paper, we define a relative homotopy theory and prove such a sequence exists.We give some immediate applications and formulate two important open questions related to the homotopy of the discrete spheres.

Definitions and notations
As usual, an undirected graph is a pair G = (V, E) where V = V(G) and E = E(G) are, respectively, the sets of vertices and edges of G.We consider only simple (i.e.without multiple edges) and connected graphs.The neighborhood (or the neighbors) of . We use the same letter N for all graphs; this will cause no confusion.We will adopt the convention that any vertex is a neighbor of itself: Recall that a graph can be defined by giving the neighbors of its vertices.We write N*(G) (or just N*) for all the neighborhoods of the graph G A pair (G,A) of graphs is just a graph G and a subgraph A. A morphism f : G → G' is an application for any .

G a 
Graphs and morphisms define a category G.A morphism of pairs of and a distinguished vertex The graph structure on We will also consider the subgraphs and, if p < q, will be written I m .
The graphs □, □ and I m are based by 0. A sequence in G is a morphism . As in topology, we have the following result.
Proposition 1.The application Proof.The necessary part is obvious.We prove the converse by contradiction.Assuming f is not a morphism, then there exists Without restricting the generality, we can and will assume that a convergent sequence : for some ; m we call it a path of length m in G and write it and , or and for the set of all of them.We can assume that any finite number of n-spheroids are defined on the same , n m I by extending them outside their domain by the constant value x 0 .If are two n-spheroids, they are homotopic, written Homotopy is an equivalence relation on  .
The following results have been proved in Benayat and Kadri (1997) and Babson et al. (2006).
Proposition 2. 1) We have the isomorphism: be a based pair of graphs.For n ≥ 1, we put: Let us recall that any finite number of nspheroids can and will be defined on the same n m I which will be written be two n-spheroids.They are homotopic if there is a morphism Homotopy of relative spheroids is an equivalence Let J n be the domain shown underneath where the second square is actually a translated copy of n I .The base J n-1 is just the juxtaposition of I n-1 and a translation of itself.Proof.The associativity of the law is easy but lengthy; we omit it.The class of the constant spheroid at 0 x is clearly the identity element.Let us show that every class has an inverse.We assume We have defined by: . The application H is a morphism since all parts of its definition glue together.Moreover, for t = 0, we have and, for t = m, we have where , ,..., n q q q q  .For all , , 1 are given in term of values of  and, consequently, define an n-spheroid of   .
, , So is homotopic to the constant n-spheroid.
We show that The following displacements of the domains I n and J n in the hyperplane Points of the empty space are sent to x 0 .This has been possible only because of the extra degree of freedom allowing rotation around the axis 0 : is in bijection with the (path) connected components of G.
The homotopy exact sequence be the obvious inclusions which are morphisms.By functoriality of the n  we get homomorphisms We define a boundary operator We get a long homotopy sequence: 2) 3) The map we consider an element is nullhomotopic and let be such a homotopy.So   ., , is precisely y and the rest of the border of p n I I  is sent to x 0 .We have gotten an element in is homotopic, in A to the constant loop x 0 and let such a homotopy where, for convenience, we take ,..., , ..., ,..., , ,..., , , ,..., , 0 0 H q q t s p p s t p K q q s t H q q t p s ,..., , ,..., , , ,..., , 0 0 H q q t s p p s t p K q q s t H q q t p s For each value of , t the application . For , 0  t we start with 0  and we end up, for Proof.These are direct consequences of the exact sequence.
Proposition 4. The image of the homomorphism  given by: where, for any k, A k is contractible and  be an n-spheroid at x 0 .Then, there is The discrete n-sphere The circle with m vertices is the quotient graph . In particular, they are contractible and have trivial homotopy.All the complete graphs are trivial.So the minimal non-trivial circle is .
of two contractible subgraphs which are the neighborhoods of the 'opposite' poles N and S, and whose intersection is G.
We define the discrete n-sphere as  and North and South poles goes to the same named vertices respectively.
It is clearly a morphism of graphs Proposition 7. We have a canonical homorphism of suspension: Proof.The suspension homomorphism is the result of the following compositions: The first isomorphism comes from the HES (homotopy exact sequence) applied to the pair In particular, we have homomorphisms of suspension .The number of vertices and edges of S n is 2 (n + 1) and 2n (n + 1) respectively.Applying the HES (homotopy exact sequence) to the pair   , S S and using results from Benayat and Kadri (1997), we get the exact sequence: .This leads to the following future work: 1) Use technology to prove the latter isomorphism.2) We have defined (not yet published) a notion of discrete fibration of graphs and proved the existence of a long exact sequence.Use the topological Hopf fibration 1 3 2   S S S to construct a combinatorial version of it.This would imply, using the mentioned exact sequence that and 3) Do we have for n > 1.This would show that the discrete spheres can play the role of n-dimensional holes in a graph and that the homotopy of graphs is able to detect them.

4) Ultimately, compute the
,1 S m n      for small values of m and n, and compare them with the topological homotopy groups of spheres.
be endowed with a natural graph structure.If f, g: G → G' map whose class is 0. Acta Scientiarum.Technology Maringá, v. 35, n. 4, p. 733-738, Oct.-Dec., 2013 Using the same trick we used before, we can move the domain of  everywhere in 2 since  Let f : G → H be a morphism of graphs.Then we have a natural morphism