Applied Mathematical Sciences, Vol. 8, 2014, no. 87, 4301 - 4307 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2014.45343

Clique Cover of Graphs

Frederick S. Gella

Department of Mathematics, Institute of Arts and Sciences Far Eastern University, Manila, Philippines

Rosalio G. Artes, Jr.

Department of Mathematics and Statistics, College of Science and Mathematics Mindanao State University - Iligan Institute of Technology Andres Bonifacio Avenue, Tibanga, 9200 Iligan City, Philippines

Copyright c 2014 Frederick S. Gella and Rosalio G. Artes, Jr. This is an open access article distributed under the Creative Commons Attribution License, which permits unre- stricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract Let G be a graph. A clique of a graph G is a nonempty subset S of V (G) such that S induces a complete graph. A family of cliques of G is a clique cover of G if for every u ∈ V (G) there exists S ∈such that u ∈ S. The clique covering number of G, denoted by cc(G), is given by cc(G) = min{|| : is a clique cover G}. For a graph G, the family of singleton subsets of V (G) is a clique cover of G. Hence, the order of a graph is an upper bound for its clique covering number. The clique covering number of a graph is equal to its order if and only if it is an empty graph. The clique covering number of a graph is equal to its size if and only if it is a star. Mathematics Subject Classification: 05C30

Keywords: clique, clique cover, clique covering number

1 Introduction

Covering theory in graphs has been studied by a number of graph theorists. Various types of covering a graph such as vertex covering, edge covering, clique 4302 Frederick S. Gella and Rosalio G. Artes, Jr. covering, tree covering, forest covering, and path triple covering has been de- fined. In this study, the author focused mainly on the clique covering of a graph. Clique covering was first investigated by Pullman [6]. He presented linear algorithms for computing the minimum number of complete subgraphs needed to cover or partition the edges of any simple graph G with maximal degree less than 5. Artes [1] used the concepts of edge cover and edge covering number of graphs. He obtained the following theorem involving the edge covering number ec(G) and the clique covering number cc(G) of a graph G: For any graph G without isolated vertices, cc(G) ≤ ec(G) and if G contains no complete subgraph of order at least 3, then ec(G)=cc(G). Let G be a graph. A clique of a graph G is a nonempty subset S of V (G) such that S induces a complete graph. A family of cliques of G is a clique cover of G if for every u ∈ V (G) there exists S ∈such that u ∈ S. The clique covering number of G, denoted by cc(G), is given by

cc(G) = min{|| : is a clique cover G}.

Let G be graph without isolated vertices. An edge in G is said to cover the vertices with which it is incident. A subset U of E(G)isanedge cover of G if for each vertex v ∈ V (G) there is an edge in U which covers v. The edge covering number of G is given by

ec(G) = min{|U| : U is an edge cover of G}.

2 Preliminary Results

This section presents some basic results and observations which will be useful in establishing the clique covering number of some graphs.

Theorem 2.1 The following hold:

(a) For any graph G, the family = {{u} : u ∈ V (G)} is a clique cover of G.

(b) For any graph G without isolated vertices, the family = {{u, v} : uv ∈ E(G)} is a clique cover of G.

(c) For any graph G, cc(G) ≥ 1.

(d) If G1 and G2 are isomorphic, then cc(G1)=cc(G2).

Theorem 2.2 Let G be a graph of order n. Then 1 ≤ cc(G) ≤ n. Clique cover of graphs 4303

Proof : This result follows from Theorem 2.1(c) and Theorem 2.1(a). 

Theorem 2.3 Let G be a connected graph of order n ≥ 1. Then cc(G)=1 ∼ if and only if G = Kn. ∼ Proof :(⇐) Assume that G = Kn. By Theorem 2.1(c), cc(Kn) ≥ 1. Hence, it suffices to show that cc(Kn) ≤ 1. Clearly, the family = {V (Kn)} is a clique cover of Kn. Thus,

cc(Kn) ≤ || = |{V (Kn)}| =1.

Accordingly, cc(Kn)=1. (⇒) Assume that cc(G) = 1. Then there exists a family = {S}, where S is a clique in G. Hence, for every vertex v ∈ V (G), v ∈ S. Thus, V (G) ⊆ S. ∼ Accordingly, S = V (G) and S = V (G) = Kn. 

The following result follows from Theorem 2.3.

Corollary 2.4 Let G be a connected graph of order n. Then cc(G) ≥ 2 if and only if G is not isomorphic to Kn.

Theorem 2.5 Let G be a graph of order n.IfG1,G2,... ,Gk are the com- k ponents of G, then cc(G)= cc(Gi). i=1

n Proof : Let i be a clique cover of Gi such that |i| = cc(Gi). Clearly, i i=1 n n covers G. Hence, cc(G) ≤ |i| = cc(Gi). Suppose now that is a i=1 i=1 clique cover of G with cc(G)=||. For each i, let i = {V (Gi) ∩ S : S ∈}. n Then i is a clique cover of Gi for each i =1, 2,... ,k. Therefore, cc(Gi) ≤ i=1 n n |i| = || = cc(G). This proves that cc(G)= Gi.  i=1 i=1

The following result gives the relationship between the clique covering num- ber of a graph and the clique covering number of its subgraphs. 4304 Frederick S. Gella and Rosalio G. Artes, Jr.

Theorem 2.6 The following hold:

(a) There exists a graph G and a proper subgraph G1 of G such that cc(G1)

(b) There exists a graph G and a proper subgraph G1 of G such that cc(G1)=cc(G).

(c) There exists a graph G and a proper subgraph G1 of G such that cc(G1) >cc(G). In view of the preceding result, the following questions arise:

1. If G1 is a subgraph of G, what properties must be possessed by G1 so that cc(G1) ≥ cc(G)?

2. If G1 is a subgraph of G what properties must be possessed by G1 so that cc(G1) ≤ ccG? Theorem 2.7 and Theorem 2.8 give answers to the above questions. Theorem 2.7 If H is an induced subgraph of G, then cc(H) ≤ cc(G).

Proof : Let G be a clique cover of G such that cc(G)=|G|. Let H = {V (H) ∩ S : S ∈G}. Then H is a clique cover of H. Hence, cc(H) ≤ |H | = |G| = cc(G).  Theorem 2.8 If H is a spanning subgraph of G, then cc(H) ≥ cc(G).

Proof : Let H and G be clique covers of H and G, respectively such that cc(H)=H and cc(G)=G. Then for every clique cover H of H, H is also a clique cover of G. Hence, cc(H) ≥ cc(G). 

3 Characterization

We now characterize all graphs whose clique covering number equal their cor- responding sizes and orders. We prove our result using the following lemmas and theorem.

Lemma 3.1 Let G be a graph of order n ≥ 2.IfG contains K2 as a subgraph, then cc(G)

Proof : Let G be a graph of order n ≥ 2 andK2 as a subgraph of G. Since V (K2) is a clique, the family = {V (K2)} {{u} : u ∈ V (G) \ V (K2)} is a clique cover of G. Hence, cc(G) ≤ || = |V (G)|−1 < |V (G)|. Therefore, cc(G) ≤ || = n − 1

Lemma 3.2 Let G be a graph of order n.Ifcc(G)=n and v ∈ V (G), then degG(v)=0.

Proof : Suppose degG(v) = 0. Then there exists u ∈ V (G) such that v is adjacent u. Let V (K2)={u, v}. Since G contains K2 as subgraph, then by Lemma 3.1 cc(G)

Theorem 3.3 Let G be a graph of order n. Then cc(G)=n if and only if ∼ G = Kn. ∼ Proof : Assume G = Kn. By Theorem 2.3 and Theorem 2.5, cc(G)=cc(Kn)= n cc(K1)=n. i=1 For the converse, suppose that cc(G)=n. By Lemma 3.2, G is Kn. 

Lemma 3.4 Let G be a connected graph of order n ≥ 4 and size m.IfG contains K3 or P4 as a subgraph, then cc(G)

Proof : Let G be a connected graph of order n ≥ 4 and K3 a subgraph of G. Let V (K3)={a, b, c}. Since b is incident to ab and c is incident to ac, it follows from the connectedness of G that the family = {{u, v} : uv ∈ E(G)}\ {{b, c}}} is a clique cover of G. Hence cc(G) ≤ || = |E(G)|−1 < |E(G)| by definition of a clique covering number of a graph. Let P4 =[a, b, c, d]. Since b is incident to ab and c is incident to cd, it follows from the connectedness of G that the family = {{u, v} : uv ∈ E(G)}\ {{b, c}} is a clique cover of G. Hence cc(G) ≤ || = m − 1

Lemma 3.5 Let G be a nontrivial connected graph of size m.Ifcc(G)=m and u, v ∈ V (G) where u = v, then either dG(u, v)=1or dG(u, v)=2.

Proof : The conclusion is clear for 1 < |V (G)|≤3. So, assume that |V (G)|≥4. Let u, v ∈ V (G) and suppose that dG(u, v) = 1 and dG(u, v) = 2. Then there exist vertices a, b ∈ V (G) such that dG(a, b) = 3. This implies that G has a subgraph H isomorphic to P4. Hence, cc(G) < |E(G)| by Lemma 3.4. This clearly contradicts our assumption. Therefore, either dG(u, v)=1or dG(u, v)=2. 

Lemma 3.6 Let G be a nontrivial connected graph of size m.Ifcc(G)=m, then there exists a unique vertex u0 of G such that u0v ∈ E(G) for every v ∈ V (G) \{u0}. Moreover, if x and y are distinct vertices of G such that x = u0 and y = u0, then dG(x, y)=2. 4306 Frederick S. Gella and Rosalio G. Artes, Jr.

Proof : The conclusion is clear for |V (G)|≤3. So, assume that |V (G)|≥4. Let x, y ∈ V (G) such that dG(x, y) = 1. Let z ∈ V (G) \{x, y}. Suppose dG(x, z) = 1 and dG(y, z) = 1. Then {x, y, z} induces a subgraph K3.By Lemma 3.4, cc(G)

Theorem 3.7 Let G be a nontrivial connected graph of size m. Then cc(G)=m if and only if G is the star K1,m. ∼ Proof : Assume G = K1,m. Since G contains no complete subgraph of order at least 3, cc(G)=ec(G)=m. For the converse, suppose that cc(G)=m. By Lemma 3.6, G is a star. In particular, G = K1,m. 

Acknowledgements. The authors thank the anonymous referees for their helpful suggestions and comments.

References

[1] R.G. Artes, Jr., On the Edge Cover of Graphs, Master’s Thesis, MSU-IIT, 2004. [2] J.A. Bondy and U.S.R. Murty, Graph Theory with Applications, Macmil- lan Press, London, 1977. Clique cover of graphs 4307

[3] G. Chartrand and L. Lesniak, Graphs & Digraphs, Chapman and Hall, New York, 1996.

[4] G. Chartrand and O.R. Oellermann, Applied and Algorithmic Graph The- ory, McGraw-Hill, Inc., New York, 1993.

[5] R.P. Gupta, Independence and covering numbers of line graphs and total graphs. New York: Academic Press, 1969.

[6] N. Pullman, Clique Covering of Graphs IV. Algorithms, http://locus.siam.org/SICOMP/volume-13/art0213005.html

Received: May 15, 2014