On Intersection Representations and Clique Partitions of Graphs

A multifamily set representation of a finite simple graph $G$ is a multifamily $\mathcal{F}$ of sets (not necessarily distinct) for which each set represents a vertex in $G$ and two sets in $\mathcal{F}$ intersects if and only if the two corresponding vertices are adjacent. For a graph $G$, an $\mathit{\text{edge}}\mathit{\text{clique}}\mathit{\text{covering}}$ ($\mathit{\text{edge}}\mathit{\text{clique}}\mathit{\text{partition}}$, respectively) $\mathcal{Q}$ is a set of cliques for which every edge is contained in $\mathit{\text{at}}\mathit{\text{least}}$ ($\mathit{\text{exactly}}$, respectively) one member of $\mathcal{Q}$. In 1966, P. Erd"{o}s, A. Goodman, and L. P'{o}sa (The representation of a graph by set intersections, $\mathit{\text{Canadian}}\mathit{\text{J.}}\mathit{\text{Math.}}$, $\mathbf{\text{18}}$, pp.106-112) pointed out that for a graph there is a one-to-one correspondence between multifamily set representations $\mathcal{F}$ and clique coverings $\mathcal{Q}$ for the edge set. Furthermore, for a graph one may similarly have a one-to-one correspondence between particular multifamily set representations with intersection size at most one and clique partitions of the edge set. In 1990, S. McGuinness and R. Rees (On the number of distinct minimal clique partitions and clique covers of a line graph, $\mathit{\text{Discrete}}\mathit{\text{Math.}}$ $\mathbf{\text{83}}$ (1990) 49-62.) calculated the number of distinct clique partitions for line graphs. In this paper, we study the set representations of graphs corresponding to edge clique partitions in various senses, namely family representations of $\mathit{\text{distinct}}$ sets, antichain representations of $\mathit{\text{mutually}}\mathit{\text{exclusive}}$ sets, and uniform representations of sets with the $\mathit{\text{same}}\mathit{\text{cardinality}}$. Among others, we completely determine the number of distinct family representations and the number of antichain representations of line graphs.