# Edge clique cover of a graph with restriction on how many times an edge can be covered

An edge clique cover of an undirected graph $G$ is a set of cliques such that every edge of $G$ belongs to some clique in the set. The edge clique cover number $\theta(G)$ is the minimum size of edge clique cover of $G$.

Let's define $k$-restricted edge clique cover of $G$ as a set of cliques such that every edge of $G$ belongs to at least one but not more the $k$ cliques in the set. The $k$-restricted edge clique cover number $\theta_k(G)$ is the minimum size of $k$-restricted edge clique cover of $G$.

My questions are following: is anything known about this concept? I'm most interested in upper bound on $\theta_k(G)$ in terms of $\theta(G)$, but any reference would help.

In general there can be a rather large difference in these quantities. If you take $k = 1$ you are considering the clique partition vs. clique covering problem. This is studied in Clique partitions and clique coverings by Erdős, Faudree, and Ordman. In Example 1 they construct a graph $G$ on $n$ vertices such that (in your notation) $\theta_1(G)/\theta(G) > n^2 /64.$ In Proposition 1 it is shown that for a graph $G$ on $n$ vertices $\theta_1(G)/\theta(G) \leq n^2/12$ provided $n$ is large. Finding an optimal constant here appears to be an open problem. See Problem (66) in Chung's Open problems of Paul Erdős in graph theory.
I am unaware of work on values of $k > 1,$ but perhaps some of the constructions in the linked paper can be applied to these cases.