Solution to a problem of Katona on counting cliques of weighted graphs

Abstract

A subset I of the vertex set V(G) of a graph G is called a k-clique independent set ofG if no k vertices in I form a k-clique of G. An independent set is a 2-clique independent set. Let πk(G) denote the number of k-cliques of G. For a function w:V(G)→{0,1,2,…}, let G(w) be the graph obtained from G by replacing each vertex v by a w(v)-clique Kv and making each vertex of Ku adjacent to each vertex of Kv for each edge {u,v} of G. For an integer m≥1, consider any w with ∑v∈V(G)w(v)=m. For U⊆V(G), we say that w is uniform onU if w(v)=0 for each v∈V(G)∖U and, for each u∈U, w(u)=m/|U| or w(u)=m/|U|. Katona asked if πk(G(w)) is smallest when w is uniform on a largest k-clique independent set of G. He placed particular emphasis on the Sperner graph Bn, given by V(Bn)={X:X⊆{1,…,n}} and E(Bn)={{X,Y}:X⊊Y∈V(Bn)}. He provided an affirmative answer for k=2 (and any G). We determine graphs for which the answer is negative for every k≥3. These include Bn for n≥2. Generalizing Sperner’s Theorem and a recent result of Qian, Engel and Xu, we show that πk(Bn(w)) is smallest when w is uniform on a largest independent set of Bn. We also show that the same holds for complete multipartite graphs and chordal graphs. We show that this is not true of every graph, using a deep result of Bohman on triangle-free graphs.