The Complexity of HCP in Digraps with Degree Bound Two

Let us named a simple strong connected digraphs with at most indegree 1 or 2 and outdegree 2 or 1 as Γ digraphs. This paper solves the HCP of Γ digraphs with following main results. Theorem 1. Given an incidence matrix C nm of Γ digraph, building a mapping:F = C + -C -, then F is a incidence matrix of undirected balanced bipartite graph G(X, Y ; E), which obeys the following properties:

c3. G has at most n 4 components which is length of 4.

Let us named the undirected balanced bipartite graph G(X, Y : E) of Γ digraph as projector graph.

Theorem 2. Let G be the projector graph of a Γ graph D(V, A), determining a Hamiltonian cycle in Γ digraph is equivalent to find a perfect match M in G and r(C ′ ) = n -1, where C ′ is the incidence matrix of D ′ (V, L) ⊆ D and L = {a i |a i ∈ D ∧ e i ∈ M }.

Let the each component of G corresponding to a boolean variable, a monotonic function f (M ) is build to represents the number of component in D. Based on this function, the maximum number of non-isomorphism perfect matching is linear, thus complexity of Γ digraphs has a answer. Theorem 3. Given the incidence matrix C nm of a Γ digraph , the complexity of finding a Hamiltonian cycle existing or not is O(n 4 )

The concepts of cycle and rank of graph are given in section 2. Then theorems 1,2,3 are proved in sections 3,4,5 respectively. The last section discusses the P versus N P in more detail.

Throughout this paper we consider the finite simple (un)directed graph D = (V, A) (G(V, E), respectively), i.e. the graph has no multi-arcs and no self loops. Let n and m denote the number of vertices V and arcs A (edges E, respectively), respectively.

As conventional, let |S| denote the number of a set S. The set of vertices V and set of arcs of A of a digraph D(V, A) are denoted by

Let the out degree of vertex v i denoted by d + (v i ), which has the in degree by denoted as d -(v i ) and has the degree

Let us define a forward relation ⊲⊳ between two arcs as following,

A cycle L is a set of arcs (a 1 , a 2 , . . . , a l ) in a digraph D, which obeys two conditions:

If a cycle L obeys the following conditions, it is a simple cycle. c3. ∀L ′ ⊂ L, L ′ does not satisfy both conditions c1 and c2.

A Hamiltonian cycle L is also a simple cycle of length n = |V | ≥ 2 in digraph. As for simplify, this paper given a sufficient condition of Hamiltonian cycle in digraph.

A graph that has at least one Hamiltonian cycle is called a Hamiltonian graph. A graph G=(V ; E) is bipartite if the vertex set V can be partitioned into two sets X and Y (the bipartition) such that ∃e i ∈ E, x j ∈ X, ∀x k ∈ X \ {x j }, (e i ⊲⊳ x j = ∅ → e i ⊲⊳ x k = ∅) (e i , Y , respectively). if |X| = |Y |, We call that G is a balanced bipartite graph. A matching M ⊆ E is a collection of edges such that every vertex of V is incident to at most one edge of M , a matching of balanced bipartite graph is perfect if |M | = |X|. Hopcroft and Karp shows that constructs a perfect matching of bipartite in O((m + n) √ n) [3]. The matching of bipartite has a relation with neighborhood of X.

Two matrices representation for graphs are defined as follows.

It is obvious that every column of an incidence matrix has exactly two 1 entries.

It is obvious to obtain a corollary of the incidence matrix as following.

Theorem 5. [5] The C is the incidence matrix of a directed graph with k components the rank of C is given by

In order to convince to describe the graph D properties, in this paper, we denotes the r(D) = r(C).

Firstly, let us divided the matrix of C into two groups.

It is obvious that the matrix of C + represents the forward arc of a digraph and C -matrix represents the backward arc respectively. A corollary is deduced as following.

Secondly, let us combined the the C + and C -as following matrix.

In more additional, let F represents as an incidence matrix of undirected graph G(X, Y ; E). The F is named as projector incidence matrix of C and G is named as projector graph , where X represents the vertices V + of D, Y represents the vertices of V -respectively. In another words we build a mapping F : D → G and denotes it as G = F (D). So the F (D) has 2n vertices and m edges if D has n vertices and m arcs. We also build up a reverse mapping: F -1 : G → D When G is a projector graph. To simplify, we also denotes the arcs a i = F -1 (e i ), v + i = F -1 (x i ) and v - i = F -1 (y i ).

Firstly, let us prove the theorem 1.

Proof. c1. Since Γ digraph is strong connected, then each vertices of Γ digraph has at least one forward arcs, each row of C + has at least one 1 entries, and the U represents the C + , so

the same principle of C -, each row of C -has at least one -1 entries, an

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