Salt, Junctions and the Long Way Round: Graphing Leicester's Winter Gritting

The first proper cold snap of winter 2026 has a way of making Leicester feel like one big question mark. Will the A6 be slushy by the school run? Is Belgrave Road passable before the morning buses? And why does the gritter sometimes seem to go past your street twice while ignoring the one next door?

Some of that is weather. Some of it is timing. But a surprising amount is maths — specifically, graph theory. A city's gritting network is a network in the literal sense, and the people who plan winter routes are solving a problem that mathematicians have been chewing over for centuries.

A city is a graph in disguise

Strip away the shop fronts, the traffic lights and the slightly optimistic parking, and a road network becomes a graph. Junctions are nodes. The stretches of road between them are edges. In Leicester's winter gritting plan, not every street is an edge that matters: the council prioritises key routes — main roads, bus corridors, access to hospitals and emergency services, steep bits and busy junctions. That smaller, high-priority web is the graph a route planner actually cares about.

Try it on a simplified Leicester map. Put a node at the Clock Tower, one at Belgrave Circle, one where Welford Road meets the ring road, one near Saffron Lane, one on Hinckley Road, one on Melton Road, one on Groby Road and one on Aylestone Road. Join them with the priority roads you'd expect a gritter to cover. You have just built a toy version of a real winter operations map — and already it is more interesting than it looks.

The odd-degree problem

Count the number of edges meeting at each node. That number is the node's degree. If every node has an even degree, something lovely happens: you can draw a route that uses every edge exactly once and returns to where you started. That is an Euler circuit, named after Leonhard Euler, who sorted out the Bridges of Königsberg in 1736. For a gritter, an Euler circuit would be the dream — every priority road salted once, no wasted repeats, back at the depot for a brew.

Real cities are rarely that polite. Leicester's simplified map will almost certainly have some odd-degree junctions. And here is the sting: a graph with more than two odd-degree nodes cannot have an Euler circuit. A route that covers every edge exactly once is impossible. The gritter must repeat some roads.

The question shifts from “Can we avoid repeats?” to “Which repeats cost the least?” That is where the Chinese postman problem comes in — also called the route inspection problem. You find the odd-degree nodes, pair them up, and work out the shortest path between each pair. Those shortest paths become the extra road you grit again. Pair them cleverly and you minimise the total distance travelled.

From Königsberg to Leicester's ring road

Imagine four odd junctions on our toy map: Belgrave Circle, the Welford Road ring-road junction, Saffron Lane and Groby Road. One pairing might send the truck along a short link between two of them but force a long detour between the other two. A different pairing might balance the repeats more neatly. The maths is not about finding a perfect route that doesn't exist; it is about choosing the least wasteful imperfect one.

This is genuinely how route planners think, even when they are using software rather than a whiteboard. The computer is doing the same job at a larger scale: nodes, edges, distances, odd vertices, shortest paths. The maths does not know it is in Leicester. It just sees a graph.

More than one truck, more than one problem

Real gritting adds complications that a single Euler trail cannot handle. Leicester does not send out one heroic truck to cover the whole priority network. It sends a fleet, each with a depot, a salt capacity, a driver shift and a fuel limit. Now the problem becomes vehicle routing: split the edges between trucks, keep each route sensible, avoid sending two gritters down the same street at once, and make sure the most critical roads are done early.

That is a much harder puzzle. But the graph-thinking is the same. You still want to avoid pointless repeats. You still want to pair odd nodes sensibly. You still want to know which roads are truly essential and which can wait until the main network is safe.

There is an environmental angle too. Every extra kilometre means more diesel, more emissions and more time before the next street gets treated. A route that saves ten minutes per truck across a cold snap is not just tidy maths — it is salt, fuel and sleep saved.

A classroom challenge: design the salt-truck route

Give students a simplified Leicester map with eight junctions and about twelve priority edges. Ask them to:

1. Label each junction with its degree.
2. Identify the odd-degree nodes.
3. Decide whether an Euler circuit is possible.
4. Pair the odd nodes and add the shortest repeat paths.
5. Write a route that covers every edge at least once, starting and finishing at the depot.
6. Compare routes with another group and argue which is better — and by what measure.

Then add a twist. Give the truck a salt limit and a ninety-minute shift. Which edges get dropped first? Which repeats become unacceptable? Suddenly the puzzle has a real-world trade-off, and students are doing operations research with a Leicester accent.

Grey Novembers will keep arriving. The forecast will keep changing. But beneath the grit and the headlights is a graph, and graphs are there to be solved. Next time a gritter rumbles past, have a look at the junction it just left. There may be an odd-degree node behind you.