_{1}

^{*}

In this paper we study the structure of
*k*-transitive closures of directed paths and formulate several properties. Concept of
*k*-transitive orientation generalizes the traditional concept of transitive orientation of a graph.

We use the standard notation. By an edge we mean an unoriented pair of vertices, and by an arc we mean an oriented pair of vertices. For a given graph G,

Orientation of a graph G is called transitive if for every

A digraph is called k-transitive if every directed path of the length k has a shortcut joining the beginning and the end of this path. In other words, if

Note that our term “k-transitive” coresponds to “

A k-transitive closure of an oriented graph

(3)

(4) it has the minimal (by inclusion) set of arcs among all graphs with the above stated properties.

Observe that there are oriented graphs for which the k-transitive closure does not exist. For example in a cyclically oriented cycle

If the k-transitive closure does exist for some oriented graph, it is unique.

Note that this definition is a partial answer to the point (4) in ([

The aim of this paper is to describe k-transitive closures of directed paths.

Instead of

Although the graph

In this paper by a degree sequence of a graph

Observe that

The starting point in a construction of the k-transitive closure of the path

The key observations are:

3.1 Fact. Adding one vertex to the path adds only arcs ending in this new vertex. In other words,

3.2 Fact. In the graph

Proof. It follows directly from the construction described above that

To show the other inclusion we use the induction on n. First observe that for

In

From the observations mentioned above, we conclude several properties of graphs

4.1 Fact. For

We can observe the following block structure in indegree/outdegree sequences of graphs

4.2 Theorem. Let

Similarly, the outdegree sequence is built from uniform “blocks” of length

Proof. The proof follows from Facts 3.2 and 3.1. We prove the part concerning the indegree sequence. First note that the indegree of the first vertex is 0. For the next

than the arcs in the initial path, so their indegree is 1. First vertex of indegree 2 is the

The proof for the outdegree sequence is similiar; we just start from the last vertex. □

4.3 Corollary. The graph

Proof. This is a consequence of Theorem 4.2; just observe that summing up the indegree and outdegree sequences gives the constant sequence

4.4 Corollary. For every

repeated to get the sequence of the length

Proof. This is another consequence of Theorem 4.2. □

4.5 Corollary. For every

As an example, below are the degree sequences for 5-transitive closures of the paths on 10, 11, 12, 13 and 14 vertices:

• for

• for

• for

• for

• for

Recall that by degree of a vertex v in a digraph we mean a pair

For oriented graphs

4.6 Corollary. Every constant subsequence in the degree sequence of the non regular graph

For example, the degree sequence for

Recall that an oriented graph G is irregular if for every two vertices

Straightforward consequence of Corollary 4.6 is that graphs

4.7 Theorem. Oriented graphs

Proof. By Theorem 4.2, pairs of vertices

Also by Theorem 4.2,

Recall that the tournament

By density of the graph G,

Recall then for every even n, a graph

For every odd n, in a graph

Observe that in both cases the density is bigger then 1/2 and

We have the following:

5.1 Theorem. For

Proof. By Corollary 4.5,

By standard calculation we get

Recall that

Obviously, for

The main open problem concerning k-transitive closures in general, is to state what properties of an oriented graph G guarantee the existence of

There are also some other special classes of oriented graphs, such as cycles (with different orientations) and trees, for which there is a chance to obtain interested properties for their k-transitive closures.

We acknowledge the support by the UJK grant No. 612439.

Some of the results contained in this paper were presented at the 5th Polish Combinatorial Conference, Będlewo, September 22-26, 2014. The author wants to express his thanks to Professor Zsolt Tuza for pointing to valuable references.

KrzysztofPszczoła, (2015) On k-Transitive Closures of Directed Paths. Advances in Pure Mathematics,05,733-737. doi: 10.4236/apm.2015.512066