A Six-Vertex Counterexample to Vertex-Averaged Ordered Reachability

We give two loopless directed graph layers on six vertices, each with constant outdegree two, having no rainbow directed cycle and hence no increasing rainbow cycle. For the fixed label order \(1<2\), the sizes of the sets reachable by possibly trivial increasing paths are \(5,5,5,5,5,4\). Their average is \(29/6<5=1+\delta_1+\delta_2\). Thus the uniform-start-vertex interpretation of the ordered-reachability conjecture attributed to DeVos in Sullivan's 2006 survey is false. Uniform blow-ups give examples on \(6m\) vertices with average \(1+23m/6\), below the proposed bound \(1+4m\) for every \(m\geq1\). The proof is an explicit neighborhood calculation and does not depend on the numerical search that located the example.