  A k-clique covering of a simple graph G, is an edge covering of G by its
cliques such that each vertex is contained in at most k cliques. The smallest k
for which G admits a k-clique covering is called local clique cover number of G
and is denoted by $lcc(G)$. Local clique cover number can be viewed as the
local counterpart of the clique cover number which is equal to the minimum
total number of cliques covering all edges. In this paper, several aspects of
the problem are studied and its relationships to other well-known problems are
discussed. Moreover, the local clique cover number of claw-free graphs and its
subclasses are notably investigated. In particular, it is proved that local
clique cover number of every claw-free graph is at most $c\Delta / \log\Delta$,
where $\Delta$ is the maximum degree of the graph and $c$ is a universal
constant. It is also shown that the bound is tight, up to a constant factor.
Furthermore, it is established that local clique number of the linear interval
graphs is bounded by $\log\Delta + 1/2 \log \log\Delta + O(1)$. Finally, as a
by-product, a new Bollobas-type inequality is obtained for the intersecting
pairs of set systems.
