On the road coloring problem

Friedman, Joel

doi:10.1090/S0002-9939-1990-0953004-8

On the road coloring problem
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by Joel Friedman PDF

Proc. Amer. Math. Soc. 110 (1990), 1133-1135 Request permission

Abstract:

Let $G = (V,E)$ be a strongly connected, aperiodic, directed graph having outdegree 2 at each vertex. A red-blue coloring of $G$ is a coloring of the edges with the colors red and blue such that each vertex has one red edge and one blue edge leaving it. Given such a coloring, we define $R:V \to V$ by $R(v) = w$ iff there is a red edge from $v$ to $w$. Similarly we define $B:V \to V$ . $G$ is said to be collapsible if some composition of $R$’s and $B$’s maps $V$ to a single vertex. The road coloring problem is to determine whether $G$ has a collapsible coloring. It has been conjectured that all such $G$ have a collapsible coloring. Since $G$ has outdegree 2 everywhere and is strongly connected, the adjacency matrix, $A$, of $G$ has a positive left eigenvector $w = (w({v_1}), \ldots ,w({v_n}))$ with eigenvalue 2 , i.e. $wA = 2w$. Furthermore, we can assume that $w$’s components are integers with no common factor. We call $w(v)$ the weight of $v$. Let $W \equiv {\Sigma _{v \in V}}w(v)$, defined to be the weight of the graph. We will prove that if $G$ has a simple cycle of length relatively prime to $W$, then $G$ is collapsibly colorable.

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