The Competition Numbers of Johnson Graphs with Diameter Four

In 2010, Kim, Park and Sano studied the competition numbers of Johnson graphs. They gave the competition numbers of J(n,2) and J(n,3).In this note, we consider the competition number of J(n,4).


Introduction
The notion of a competition graph was introduced by Cohen [1] as a means of determining the smallest dimension of ecological phase space. The competition graph C(D) of a digraph D is a simple undirected graph which has the same vertex set as D and an edge between vertices x and y if and only if there exists a vertex u ∈ D such that (x, u) and (y, u) are arcs of D. For any graph G, G together with sufficiently many isolated vertices is the competition graph of an acyclic digraph. Roberts [7] defined the competition number k(G) of a graph G to be the smallest number k such that G together with k isolated vertices is the competition graph of an acyclic digraph. Opsut [4] showed that the computation of the competition number of a graph is an NP-hard problem. In the study of competition graphs, it has been one of important problems to determine the competition numbers for various graph classes. In [3], Kim, Park and Sano studied the competition numbers of Johnson graphs. In particular, they gave the following results. www.ejgta.org The competition numbers of Johnson graphs with diameter four | Kijung Kim Theorem 1.1 (See [3]). For n ≥ 4, we have k(J(n, 2)) = 2. Theorem 1.2 (See [3]). For n ≥ 6, we have k(J(n, 3)) = 4.
They also asked about the exact value of the competition number of J(n, 4). In this note, we give a partial answer to the question. Our result is the following. Theorem 1.3. For n ≥ 8, we have k(J(n, 4)) ∈ {7, 8, 9}.

Preliminaries
Throughout this note, we use the notations given in [3]. We denote an n-set {1, . . . , n} by [n] and the set of all d-subsets of n-set by [n] d . The Johnson graph J(n, d) is an undirected graph whose vertex set is {v X | X ∈ [n] d }, and two vertices v X 1 and v X 2 are adjacent if and only if It is well known that a digraph D is acyclic if and only if there exists an acyclic ordering of D.
For a digraph D and a vertex v of D, we define the out-neighborhood We also use N G (v) to stand for the subgraph induced by its vertices.
For a clique S of a graph G and an edge e of G, we say e is covered by S if both of the endpoints of e are contained in S. An edge clique cover of a graph G is a family of cliques such that each edge of G is covered by some clique in the family. The edge clique cover number θ E (G) of a graph G is the minimum size of an edge clique cover of G. An edge clique cover of G is called a minimum edge clique cover of G if its size is equal to θ E (G). A vertex clique cover of a graph G is a family of cliques such that each vertex of G is contained in some clique in the family. The vertex clique cover number θ V (G) of a graph G is the minimum size of a vertex clique cover of G.
A minimum edge clique cover of J(n, d) is given in [3] as follows.
d−1 } is the collection of cliques of maximum size. We denote it by F n d . Note that F n d is an edge clique cover of J(n, d). Lemma 2.1 (See Section 3 of [3]). We have θ E (J(n, d)) = n d−1 , and F n d is a minimum edge clique cover of J(n, d).

Main results
In this section, we give a lower bound for the competition number of J(n, d) and an upper bound for the competition number of J(n, 4).
Proof. We denote k(J(n, d)) by k. Then there exists an acyclic digraph D such that C(D) = J(n, d) ∪ I k , where I k = {z 1 , z 2 , . . . , z k } is a set of isolated vertices. Let x 1 , x 2 , . . . , x ( n d ) , z 1 , z 2 , . . . , z k be an acyclic ordering of D.
Since x 1 , x 2 , . . . , x ( n d ) , z 1 , z 2 , . . . , z k is an acyclic ordering of D, we have First of all, we assume that v 1 and v 2 are not adjacent in J(n, d).
Then v 1 and v 2 do not have a common prey in D, that is, It follows from (1), (2) and (3) that So, we have k ≥ 2d − 1.
Next, we assume that v 1 and v 2 are adjacent in J(n, d).
Then v 1 and v 2 have at least one common prey in D, that is, Now we divide our consideration into four cases: 1. In the first case, we have (1), Lemma 3.2 and (4)) = 3d − 1.
In the second case, we have So, we have k ≥ 3d − 4.
In the third case, we have So, we have k ≥ 3d − 4.
In the fourth case, we have So, we have k ≥ 3d − 5.
Now we give an order ≺ on the vertex set of J(n, d) as follows. Take two distinct elements v X 1 and v X 2 in {v X | X ∈ [n] d }. Let X 1 = {i 1 , . . . , i d } and X 2 = {j 1 , . . . , j d }, where i 1 < · · · < i d and j 1 < · · · < j d . Then we define v X 1 ≺ v X 2 if there exists t ∈ {1, . . . , d} such that i s = j s for 1 ≤ s ≤ t − 1 and i t < j t .
Proof. We define a digraph D as follows: where I 9 = {z 1 , . . . , z 9 }, and It is easy to see that