Elimination pattern conjecture for admissible colorings

Let $K$ be a combinatoric, let $g$ be the genus of $\abs{K}$, and let $K^\ast$ be its dual complex. An admissible coloring of $K^\ast$ is a coloring satisfying the condition that every two-simplex contains exactly three arrows pointing outward or corners. Elimination pattern conjecture. There is an admissible coloring of $K^\ast$ such that, if $g=0$, every one-simplex of $K^\ast$ has exactly one arrow, exactly six corners of $K^\ast$ are colored, and for any two adjacent zero-simplices $i$ and $j$ the colored corners can be chosen as the union of the corners of $i$ and $j$; while, if $g\geq 1$, exactly one arrow lies on all but $6g-6$ one-simplices of $K^\ast$, and no corners are colored. This coloring is intended to provide the pattern needed for the proof of the paper's main theorem; whether it exists in the stated generality is left unresolved in the supplied text.