On Greedy Clique Decompositions and Set Representations of Graphs

TL;DR

This paper explores greedy clique decompositions and their relation to set representations of graphs, establishing bounds on the number of cliques in such decompositions and their connection to set family representations.

Contribution

It extends McGuinness's result by analyzing greedy clique decompositions through the lens of set representations with distinct sets, confirming the same clique bound.

Findings

Greedy clique decompositions have at most  cliques for n-vertex graphs.

A correspondence exists between certain set representations and clique partitions.

The bound applies to greedy methods using distinct set families.

Abstract

In 1994 S. McGuinness showed that any greedy clique decompo- sition of an n-vertex graph has at most $\lfloor n^2/4 \rfloor$ cliques (The greedy clique decomposition of a graph, J. Graph Theory 18 (1994) 427-430), where a clique decomposition means a clique partition of the edge set and a greedy clique decomposition of a graph is obtained by removing maximal cliques from a graph one by one until the graph is empty. This result solved a conjecture by P. Winkler. A multifamily set rep- resentation of a simple graph G is a family of sets, not necessarily distinct, each member of which represents a vertex in G, and the in- tersection of two sets is non-empty if and only if two corresponding vertices in G are adjacent. It is well known that for a graph G, there is a one-to-one correspondence between multifamily set representations and clique coverings of the edge set. Further for a graph one may have a one-to-one correspondence between particular multifamily set rep- resentations with intersection size at most one and clique partitions of the edge set. In this paper, we study for an n-vertex graph the variant of the set representations using a family of distinct sets, including the greedy way to get the corresponding clique partition of the edge set of the graph. Similarly, in this case, we obtain a result that any greedy clique decomposition of an n-vertex graph has at most $\lfloor n^2/4 \rfloor$ cliques.