Edge clique covering sum of graphs

The edge clique cover sum number (resp. edge clique partition sum number) of a graph G , denoted by scc( G ) (resp. scp( G )), is defined as the smallest integer k for which there exists a collection of complete subgraphs of G , covering (resp. partitioning) all edges of G such that the sum of sizes of the cliques is at most k . By definition, scc( G ) ≦ scp( G ). Also, it is known that for every graph G on n vertices, scp( G ) ≦ n 2 / 2 . In this paper, among some other results, we improve this bound for scc( G ). In particular, we prove that if G is a graph on n vertices with no isolated vertex and the maximum degree of the complement of G is d − 1, for some integer d , then scc( G ) ≦ c n d log ( n - 1 ) / ( d - 1 ) , where c is a constant. Moreover, we conjecture that this bound is best possible up to a constant factor. Using a well-known result by Bollobás on set systems, we prove that this conjecture is true at least for d = 2. Finally, we give an interpretation of this conjecture as an interesting set system problem which can be viewed as a multipartite generalization of Bollobás’ two families theorem.