The maximum number of cliques in dense graphs.

*(English)*Zbl 0569.05029We will consider only undirected, connected graphs without loops or multiple edges. Denote the number of vertices of G by \(| G|\). A clique of graph G is a maximal complete subgraph. The clique graph K(G) of G is the intersection graph of the cliques of G. The density w(G) is the number of vertices in the largest clique of G. A graph is called dense if w(G)\(\geq | G| /2.\)

This paper makes precise the intuitive idea that very dense graphs have fewer cliques than less dense graphs. First, it is shown that for any graph G, \(2^{| G| -w(G)}\geq | K(G)|.\) Secondly, this bound is sharp among dense graphs, and among them only. In fact, for all integers s,t\(\geq 4\) where \(t\geq s\geq t/2\), there exists a graph G such that \(| G| =t\), \(w(G)=s\), and \(2^{t-s}=| K(G)|.\) Call a dense graph packed if \(2^{| G| -w(G)}=| K(G)|.\) The 2n- Neumann graph is the complement of a matching between 2n vertices. Thirdly, it is shown that any packed graph G contains an induced subgraph isomorphic to the 2[\(| G| -w(G)]\)-Neumann graph. Lastly, the clique graphs of packed graphs are characterized.

This paper makes precise the intuitive idea that very dense graphs have fewer cliques than less dense graphs. First, it is shown that for any graph G, \(2^{| G| -w(G)}\geq | K(G)|.\) Secondly, this bound is sharp among dense graphs, and among them only. In fact, for all integers s,t\(\geq 4\) where \(t\geq s\geq t/2\), there exists a graph G such that \(| G| =t\), \(w(G)=s\), and \(2^{t-s}=| K(G)|.\) Call a dense graph packed if \(2^{| G| -w(G)}=| K(G)|.\) The 2n- Neumann graph is the complement of a matching between 2n vertices. Thirdly, it is shown that any packed graph G contains an induced subgraph isomorphic to the 2[\(| G| -w(G)]\)-Neumann graph. Lastly, the clique graphs of packed graphs are characterized.

##### MSC:

05C35 | Extremal problems in graph theory |

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##### References:

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