Given a graph G = (V,E) without an odd hole or anti odd hole, which we now call as Perfect (thanks to Strong Perfect Graph Theorem). My question is, suppose in such a graph we find all the maximal cliques. Suppose the clique no. is "w" which is also the chromatic no. for that graph. Now can we assign flows $x_i$ ($ 1 \leq i \leq m,\;\; 0 \leq x_i \leq 1$) to the nodes such that the clique constraints are satisfied?
i.e. $$ \sum_{i \in Q} x_i \leq 1, \; \forall \; \text{maximal cliques Q of G} $$ Just for completion, it doesn't if graph is not perfect. E.g. Pentagon graph. Here the clique constraints give all $x_i$ = 0.5. But this can't be applied on this particular graph.
I was able to proved this for a special case of graphs in which we have a linear indexing of nodes (1,2,..,N) with all the cliques having a property that: if $i$ and $j$ are in a clique $Q$ then all nodes $( k: i < k < j )$ are also in the same clique.
