On the number of distinct minimal clique partitions and clique covers of a line graph (Q1813995)

Let \(G\) be a graph. Then \(cc(G)\) \((cp(G))\), the clique covering (the clique partition) number of \(G\) is the minimum number of cliques of \(G\) that cover (partition) all edges of \textit{G. Orlin} [Nederl. Akad. Wet., Proc., Ser. A 80, Indag. Math. 39, 406-424 (1977; Zbl 0374.05041)] determined \(cc(G)\) and \(cp(G)\) in the case when \(G\) is a line graph. In this paper a constructive proof of the result is provided. Moreover, the number of distinct minimal clique covers (partitions) is given.