Is every perfect graph $G$ a union of $\omega(G)-1$ bipartite graphs?

Question

If any perfect graph $G$ has no $n+1$ clique then can one always find $n$ bipartite graphs $B_1,\ldots B_n$ such that $G=\cup_{k=1}^nB_k$?

Answer 1

Any $(n+1)$-colorable graph is the edge-disjoint union of $n$ bipartite graphs. Order the colors as $1, 2, \dots, n+1$ and let $B_k$ be the subgraph formed by all edges between color $k$ and colors $k+1, k+2, \dots, n+1$.

We can do better and use about $\log_2 n$ colors: if a graph is $2^k$-colorable, then name the colors after elements of $\{0,1\}^k$, and for $i=1, \dots, k$ let $B_i$ be the graph joining colors with a $0$ in the $i^{\text{th}}$ position to colors with a $1$ in the $i^{\text{th}}$ position.

(As stated, some edges will appear in multiple $B_i$, but if that's a problem we can make an arbitrary choice for each such edge.)