Minimal sense of direction and decision problems for Cayley graphs

Sense of direction is a property of the labelling of (possibly anonymous) networks which allows to assign coherently local identifiers to other processors on the basis of the route followed by incoming messages. A graph has minimal sense of direction whenever it has sense of direction and the number of colours equals its maximum outdegree. We prove that an outregular digraph with minimal weak sense of direction is a Cayley colour graph (in the general sense, i.e., we do not require connectedness). Since Cayley colour graphs are known to possess minimal transitive sense of direction, we obtain a characterization of outregular graphs with minimal (weak,transitive) sense of direction. As a consequence, deciding whether a coloured graph is a Cayley colour graph reduces to deciding whether it has weak sense of direction, which can be done in AC1.