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) 5 scp(G). Also, it is known that for every graph G on n vertices, scp(G) 5 n/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) 5 cnd ⌈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.