Clique coverings and partitions of line graphs

A clique   in a graph GG is a complete subgraph of GG. A clique covering (partition  ) of GG is a collection CC of cliques such that each edge of GG occurs in at least (exactly) one clique in CC. The clique covering (partition) number  cc(G)cc(G) (cp(G)cp(G)) of GG is the minimum size of a clique covering (partition) of GG. This paper gives alternative proofs, using a unified approach, for the results on the clique covering (partition) numbers of line graphs obtained by McGuinness and Rees [On the number of distinct minimal clique partitions and clique covers of a line graph, Discrete Math. 83 (1990) 49–62]. We also employ the proof techniques to give an alternative proof for the De Brujin–Erdős Theorem.