Given a graded algebra $A$, Steenrod's problem asks if $A$ can be realized. That is if there is a space $X$ such that $H^{*}(X)=A$. The Hopf invariant one problem is a famous example. Around 15 years ago Anderson-Grodal completed the solution of this problem for polynomial rings. We are interested in the case of the face ring $SR(K)$ of a simplicial complexes $K$. If all generators have degree 2 or 4 then $SR(K)$ can always be realized using polyhedral products. If there is a single generator of degree 4 then the problem has been mostly solved by Takeda. We consider the case where all generators are of degree 4 and 6. This gives rise to a graph coloring problem, and a kind of coloring which can be thought of as a localization of usual coloring.

This is joint work with Masahiro Takeda.