On Intersection Representations and Clique Partitions of Graphs

TL;DR

This paper explores set representations of graphs related to edge clique partitions, focusing on line graphs, and determines counts of specific types of representations such as family and antichain representations.

Contribution

It provides a comprehensive analysis of set representations linked to edge clique partitions, including exact counts for line graphs and various representation types.

Findings

Determined the number of distinct family representations of line graphs.

Calculated the number of antichain representations of line graphs.

Analyzed uniform set representations with fixed cardinality.

Abstract

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 \textit{edge clique covering} (\textit{edge clique partition}, respectively) $\mathcal{Q}$ is a set of cliques for which every edge is contained in \textit{at least} (\textit{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, \textit{Canadian J. Math.}, \textbf{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, \textit{Discrete Math.} \textbf{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 \textit{distinct} sets, antichain representations of \textit{mutually exclusive} sets, and uniform representations of sets with the \textit{same cardinality}. Among others, we completely determine the number of distinct family representations and the number of antichain representations of line graphs.