Graph whose smallest edge clique cover is not the collection of all maximal cliques

Let $$G$$ be an undirected simple graph. An edge clique cover of $$G$$ is a collection $$\mathcal{C}$$ of cliques (i.e., complete subgraphs) that cover all the edges of $$G$$. In other words, every edge belongs to at least one clique $$C \in \mathcal{C}$$. The smallest such collection of cliques $$\mathcal{C}$$ is sometimes known as the intersection number of $$G$$.

Clearly, we can take $$\mathcal{C}$$ to be the collection of all maximal cliques of $$G$$. This can also be optimal, i.e., there is no smaller $$\mathcal{C}$$ which is also an edge clique cover.

But is there an example of a graph such that its smallest edge clique cover is different from the collection of all maximal cliques? As a bonus question, do we know exactly when the collection of all maximal cliques is an optimal collection in terms of the number of elements in $$\mathcal{C}$$?

• The bonus question part still remains, but I suspect this might be much more difficult. – advanced_learner Dec 14 '18 at 13:30

Consider the octahedron. All $$8$$ triangular faces are maximal cliques, however only $$4$$ of them is enough for an edge clique cover.