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It is hard to compute the competition number for a graph in general and characterizing a graph by its competition number has been one of important research problems in the study of competition graphs. Sano pointed out that it would be interesting to compute the competition numbers of some triangulations of a sphere as he got the exact value of the competition numbers of regular polyhedra. In this paper, we study the competition numbers of several kinds of triangulations of a sphere, and get the exact values of the competition numbers of a 24-hedron obtained from a hexahedron by adding a vertex in each face of the hexahedron and joining the vertex added in a face with the four vertices of the face, a class of dodecahedra constructed from a hexahedron by adding a diagonal in each face of the hexahedron, and a triangulation of a sphere with 3
*n* (
*n*
≥2) vertices.

Let G = ( V , E ) be a graph in which V is the vertex set and E the edge set. We always use | V | and | E | to denote the vertex number and the edge number of G , respectively. The notion of competition graph was introduced by Cohen [

Roberts [

All graphs considered in this paper are simple and connected. For a vertex v in a graph G , let the open neighborhood of v be defined by

N G ( v ) = { u | u isadjacentto v } . For any set U of vertices in G , we define the neighborhood of U in G to be the set of all vertices adjacent to vertices in U , this set is denoted by N G ( U ) . Let N G [ U ] = N G ( U ) ∪ U and

E G [ U ] = { e ∈ E ( G ) | e hasanendpointin U } . We denote by the same symbol N G [ U ] the subgraph of G induced by N G [ U ] . Note that E G [ U ] is con- tained in the edge set of the subgraph N G [ U ] .

A subset U of the vertex set of a graph G is called a clique of G if G [ U ] is a complete graph. For a clique U of a graph G and an edge e of G , we say e is covered by U if both of the endpoints of e are contained in U . 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 . 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 .

Let G be a graph and F ⊆ E ( G ) be a subset of the edge set of G . An edge clique cover of F in G is a family of cliques of G such that each edge in F is covered by some clique in the family. The edge clique cover number θ e ( F ; G ) of F ⊆ E ( G ) in G is defined as the minimum size of an edge clique cover of F in G , i.e.,

θ e ( F ; G ) = min { | U | | U isanedgecliquecoverof F in G } . Note that the edge clique cover number θ e ( E ( G ) ; G ) of E ( G ) in a graph G is equal to the edge clique cover number θ e ( G ) of the graph G .

Opsut [

Theorem 1 (Opsut [

Theorem 2 (Opsut [

Recently, Sano [

Theorem 3 (Sano [

k ( G ) ≥ min U ∈ ( V m ) θ e ( E G [ U ] ; N G [ U ] ) − m + 1 ,

where ( V m ) denotes the set of all m -subsets of V .

The following results from [

Theorem 4 (Harary et al. [

1) For all i , j ∈ { 1 , 2 , ⋯ , n } , ( v i , v j ) ∈ A implies that i < j ;

2) For all i , j ∈ { 1 , 2 , ⋯ , n } , ( v i , v j ) ∈ A implies that i > j .

Sano [

In the following, “ S → v ” means that we make an arc from each vertex in S to the vertex v .

In this section we study the competition number of a 24-hedron obtained from a hexahedron by adding a vertex in each face of the hexahedron and joining the vertex added in a face with the four vertices of the face. See

Theorem 5. Let T be the 24-hedron shown in

Proof. Let V ( T ) = { v 1 , v 2 , ⋯ , v 14 } and suppose that the adjacencies between two vertices are given as

S 1 = { v 2 , v 7 , v 9 } , S 2 = { v 1 , v 2 , v 13 } , S 3 = { v 1 , v 4 , v 14 } , S 4 = { v 2 , v 3 , v 4 } , S 5 = { v 3 , v 6 , v 7 } , S 6 = { v 4 , v 5 , v 6 } , S 7 = { v 5 , v 10 , v 14 } , S 8 = { v 6 , v 8 , v 10 } , S 9 = { v 7 , v 8 , v 11 } , S 10 = { v 9 , v 11 , v 13 } , S 11 = { v 10 , v 11 , v 12 } , S 12 = { v 12 , v 13 , v 14 } .

Then the family { S 1 , S 2 , ⋯ , S 12 } is an edge clique cover of T .

Now, we define a digraph D as follows. Let

V ( D ) = V ( T ) ∪ { a , b , c } ,

S 1 → a , S 2 → b , S 3 → c , S 4 → v 1 , S 5 → v 2 , S 6 → v 3 , S 7 → v 4 , S 8 → v 5 , S 9 → v 6 , S 10 → v 7 , S 11 → v 8 , S 12 → v 9 ,

where a , b and c are new added vertices. Then by Theorem 4 the digraph D is acyclic and C ( D ) = T ∪ { a , b , c } . Hence we have

k ( T ) ≤ 3. (1)

On the other hand, by Theorem 3 with m = 2 , we have

k ( T ) ≥ min U ∈ ( V 2 ) θ e ( E T [ U ] ; N T [ U ] ) − 1.

There are 7 different cases for the set E T [ U ] of edges in the subgraph

N T [ U ] , where U ∈ ( V 2 ) (see

1) If T [ U ] = P 2 , then θ e ( E T [ U ] ; N T [ U ] ) = 5 ;

2) If T [ U ] = I 2 , then θ e ( E T [ U ] ; N T [ U ] ) = 6 ;

3) If T [ U ] = I 2 , then θ e ( E T [ U ] ; N T [ U ] ) = 6 ;

4) If T [ U ] = I 2 , then θ e ( E T [ U ] ; N T [ U ] ) = 4 ;

5) If T [ U ] = I 2 , then θ e ( E T [ U ] ; N T [ U ] ) = 4 ;

6) If T [ U ] = P 2 , then θ e ( E T [ U ] ; N T [ U ] ) = 4 ;

7) If T [ U ] = I 2 , then θ e ( E T [ U ] ; N T [ U ] ) = 5.

Thus it holds that

min U ∈ ( V 2 ) θ e ( E T [ U ] ; N T [ U ] ) = 4.

Hence we have

k ( T ) ≥ 3. (2)

Combining inequalities (1) and (2) we have k ( T ) = 3 .

In this section we study the competition numbers of a class of dodecahedra constructed from a hexahedron by adding a diagonal in each face of the hexahedron. It is not difficult to see that there are 6 nonisomorphic such dodecahedra. Denote the 6 different dodecahedra by ℍ 1 , ℍ 2 , ℍ 3 , ℍ 4 , ℍ 5 and ℍ 6 , respectively. See

Theorem 6. k ( ℍ i ) = { 2 , i = 1 , 3 ; 1 , i = 2 , 4 , 5 , 6.

Proof. Let V ( ℍ i ) = { v 1 , v 2 , ⋯ , v 8 } and suppose that the adjacencies between two vertices are given as

1) D 1 .

Let V ( D 1 ) = V ( ℍ 1 ) ∪ { a 1 , b 1 } , and A ( D 1 ) be defined as follows:

S 1 = { v 1 , v 2 , v 6 } → a 1 , S 2 = { v 1 , v 3 , v 5 } → b 1 , S 3 = { v 2 , v 3 , v 8 } → v 1 , S 4 = { v 3 , v 4 , v 5 } → v 2 , S 5 = { v 4 , v 7 , v 8 } → v 3 , S 6 = { v 5 , v 6 , v 7 } → v 4 , S 7 = { v 6 , v 8 } → v 5 ,

where a 1 and b 1 are new added vertices. Note that the family { S 1 , S 2 , ⋯ , S 7 } is an edge clique cover of ℍ 1 .

2) D 2 .

Let V ( D 2 ) = V ( ℍ 2 ) ∪ { a 2 } , and A ( D 2 ) be defined as follows:

S 1 = { v 1 , v 2 , v 3 , v 5 } → a 2 , S 2 = { v 2 , v 6 , v 8 } → v 1 , S 3 = { v 3 , v 4 , v 6 } → v 2 , S 4 = { v 4 , v 5 , v 7 } → v 3 , S 5 = { v 5 , v 7 , v 8 } → v 4 , S 6 = { v 6 , v 7 } → v 5 ,

where a 2 is a new added vertex. Note that the family { S 1 , S 2 , ⋯ , S 6 } is an edge clique cover of ℍ 2 .

3) D 3 .

Let V ( D 3 ) = V ( ℍ 3 ) ∪ { a 3 , b 3 } , and A ( D 3 ) be defined as follows:

S 1 = { v 1 , v 4 , v 8 } → a 3 , S 2 = { v 1 , v 2 , v 7 } → b 3 , S 3 = { v 2 , v 3 , v 4 } → v 1 , S 4 = { v 3 , v 5 , v 7 } → v 2 , S 5 = { v 4 , v 5 , v 6 } → v 3 , S 6 = { v 6 , v 7 , v 8 } → v 4 ,

where a 3 and b 3 are new added vertices. Note that the family { S 1 , S 2 , ⋯ , S 6 } is an edge clique cover of ℍ 3 .

4) D 4 .

Let V ( D 4 ) = V ( ℍ 4 ) ∪ { a 4 } , and A ( D 4 ) be defined as follows:

S 1 = { v 1 , v 3 , v 4 , v 8 } → a 4 , S 2 = { v 2 , v 4 , v 6 , v 8 } → v 1 , S 3 = { v 3 , v 5 , v 7 } → v 2 , S 4 = { v 4 , v 5 , v 6 } → v 3 , S 5 = { v 6 , v 7 , v 8 } → v 4 .

where a 4 is a new added vertex. Note that the family { S 1 , S 2 , ⋯ , S 5 } is an edge clique cover of ℍ 4 .

5) D 5 .

Let V ( D 5 ) = V ( ℍ 5 ) ∪ { a 5 } , and A ( D 5 ) be defined as follows:

S 1 = { v 1 , v 3 , v 4 , v 5 } → a 5 , S 2 = { v 2 , v 6 , v 7 , v 8 } → v 1 , S 3 = { v 3 , v 6 , v 7 } → v 2 , S 4 = { v 4 , v 6 , v 8 } → v 3 , S 5 = { v 5 , v 7 , v 8 } → v 4 ,

where a 5 is a new added vertex. Note that the family { S 1 , S 2 , ⋯ , S 5 } is an edge clique cover of ℍ 5 .

6) D 6 .

Let V ( D 6 ) = V ( ℍ 6 ) ∪ { a 6 } , and A ( D 6 ) be defined as follows:

S 1 = { v 1 , v 5 , v 6 , v 7 } → a , S 2 = { v 2 , v 5 , v 7 , v 8 } → v 1 , S 3 = { v 3 , v 5 , v 6 , v 8 } → v 2 , S 4 = { v 4 , v 6 , v 7 , v 8 } → v 3 ,

where a 6 is a new added vertex. Note that the family { S 1 , S 2 , S 3 , S 4 } is an edge clique cover of ℍ 6 .

By Theorem 4, each D i is acyclic and

C ( D i ) = { ℍ i ∪ { a i , b i } , i = 1 , 3 ; ℍ i ∪ { a i } , i = 2 , 4 , 5 , 6.

Hence we have

k ( ℍ i ) { ≤ 2 , i = 1 , 3 ; ≤ 1 , i = 2 , 4 , 5 , 6. (3)

On the other hand, we note that

・ δ ( ℍ 1 ) = δ ( ℍ 3 ) = 4 and there is no clique with more than 3 vertices in ℍ 1 and ℍ 3 , respectively;

・ N ℍ 2 ( v 1 ) = { v 2 , v 3 , v 5 } is covered by the clique S 1 = { v 1 , v 2 , v 3 , v 5 } in ℍ 2 ;

・ N ℍ 4 ( v 1 ) = { v 3 , v 4 , v 8 } is covered by the clique S 1 = { v 1 , v 3 , v 4 , v 8 } in ℍ 4 ;

・ N ℍ 5 ( v 1 ) = { v 3 , v 4 , v 5 } is covered by the clique S 1 = { v 1 , v 3 , v 4 , v 5 } in ℍ 5 ;

・ N ℍ 6 ( v 1 ) = { v 5 , v 6 , v 7 } is covered by the clique S 1 = { v 1 , v 5 , v 6 , v 7 } in ℍ 6 .

Then we have

θ v ( N ℍ i ( v ) ) ≥ 2 , v ∈ V ( ℍ i ) , where i = 1 , 3 ,

and

θ v ( N ℍ i ( v 1 ) ) = 1 , where i = 2 , 4 , 5 , 6.

By Theorem 2

k ( ℍ i ) ≥ min { θ v ( N ℍ i ( v ) ) | v ∈ V ( ℍ i ) } ≥ { 2 , i = 1 , 3 ; 1 , i = 2 , 4 , 5 , 6. (4)

Combining inequalities (3) and (4) we have

k ( ℍ i ) = { 2 , i = 1 , 3 ; 1 , i = 2 , 4 , 5 , 6.

In this section we study a graph G ( n ) = ( V , E ) with 3 n ( n ≥ 2 ) vertices, where

V = ∪ i = 1 n { x i , y i , z i }

and

E = ∪ i = 1 n { x i y i , y i z i , x i z i } ∪ ∪ i = 1 n − 1 { x i x i + 1 , x i z i + 1 , y i y i + 1 , y i z i + 1 , z i x i + 1 , z i y i + 1 } .

An example G ( 3 ) is shown in

Theorem 7. For each n ≥ 2 , k ( G ( n ) ) = 2 .

Proof. Let σ be a vertex ordering of G ( n ) such that σ ( x i ) = v 3 i − 2 ,

σ ( y i ) = v 3 i − 1 and σ ( z i ) = v 3 i , where i = 1 , 2 , ⋯ , n . Let

S 1 = { v 1 , v 2 , v 3 } , S 3 i − 1 = { v 3 i − 2 , v 3 i + 1 , v 3 i + 3 } , S 3 i = { v 3 i − 1 , v 3 i + 2 , v 3 i + 3 } and S 3 i + 1 = { v 3 i , v 3 i + 1 , v 3 i + 2 } , where i = 1 , ⋯ , n − 1.

Then the family { S 1 , S 2 , ⋯ , S 3 n − 2 } is an edge clique cover of G ( n ) . An edge clique cover of G ( 3 ) is shown in

Now, we define a digraph D by the following:

V ( D ) = V ( G ( n ) ) ∪ { a , b } , S 1 = { v 1 , v 2 , v 3 } → v 4 , S 3 i − 1 = { v 3 i − 2 , v 3 i + 1 , v 3 i + 3 } → v 3 i + 4 , S 3 i = { v 3 i − 1 , v 3 i + 2 , v 3 i + 3 } → v 3 i + 5 , and S 3 i + 1 = { v 3 i , v 3 i + 1 , v 3 i + 2 } → v 3 i + 3 , where i = 1 , ⋯ , n − 1.

Note that v 3 n + 1 = a , v 3 n + 2 = b are new vertices. Then by Theorem 4 the digraph D is acyclic and C ( D ) = G ( n ) ∪ { a , b } . Hence we have

k ( G ( n ) ) ≤ 2. (5)

On the other hand, since δ ( G ( n ) ) = 4 and there is no clique with more than 3 vertices in G ( n ) , then by Theorem 2

k ( G ( n ) ) ≥ min { θ v ( N G ( n ) ( v ) ) | v ∈ V ( G ( n ) ) } ≥ 2. (6)

Combining inequalities (5) and (6) we have k ( G ( n ) ) = 2 .

In this paper, we provide the exact values of the competition numbers of a 24-hedron, a class of dodecahedra and a triangulation of a sphere with

3 n ( n ≥ 2 ) vertices. It would be interesting to compute the competition numbers of some other triangulations of a sphere.

For a digraph D = ( V , A ) , if we partition V into k types, then we may construct a undirected graph C k ( D ) = ( V , E ) of D as follows:

1) u v ∈ E if and only if there exists some vertex x ∈ V such that

( u , x ) , ( v , x ) ∈ A and u , v are of the same type, or

2) u v ∈ E if and only if there exists some vertex x ∈ V such that

( u , x ) , ( v , x ) ∈ A and u , v are of different types.

It is easy to see that C 1 ( D ) = C ( D ) for a given digraph D , and we note that multitype graphs can be used to study the multi-species in ecology and have been deeply studied (see [

We thank the editor and the referee for their valuable comments. The work was supported in part by the Natural Science Foundation of Hebei Province of China under Grant A2015106045, and in part by the Institute of Applied Mathematics of Shijiazhuang University.

Zhao, Y.Q., Fang, Z.M., Cui, Y.G., Ye, G.Y. and Cao, Z.J. (2017) Competition Numbers of Several Kinds of Triangulations of a Sphere. Open Journal of Discrete Mathematics, 7, 54-64. https://doi.org/10.4236/ojdm.2017.72006