Cliques in $k$-partite graphs Recall that a _$k$-partite graph_ is a graph whose vertices are or can be partitioned into k different independent sets and that a _clique_ of a graph is subset of the vertices such that every two distinct vertices are adjacent. Suppose that the clique number of a graph $G$ is $ω(G)=k$ and that every vertex of $G$ belongs to a clique of size $k$. Does this imply that the graph $G$ is $k$-partite? If not, what additional conditions would guarantee this property?

"Does this imply that the graph G is k-partite?" No. The simplest example is an odd cycle. See < for examples of graphs with clique size 3 and large chromatic number.

The condition that every vertex belongs to a clique of size $k$ is not relevant as dummy vertices can be added to form such cliques.

"If not, what additional conditions would guarantee this property?" Perfect graphs such as bipartite graphs, chordal graphs etc. satisfy this property.