# Graph theory in Formula 1

This post attempts to visualize Formula 1 championships (1985-2018) by way of graphs: the outcome of each race is represented by an edge between the drivers who finished #1 and #2. The graph is undirected (no distinction between the winner and 2nd place is made), and simple (no record of multiple edges is kept). This erases some of the information, but depending on how much you care about F1, the graphs may still be enough to bring back some memories.

All graph-theoretical “records” are based on 1985-2018 data only, 2019 season being the subject of a separate post. Some highlights:

• Most vertices: 12 in 1997
• Fewest vertices: 5 in 2000 and 2011
• Most edges: 16 in 2012
• Fewest edges: 6 in 1988, 2002, 2011, and 2015
• Largest maximal degree: 6 in 1990, 1997, 2004, and 2012
• Smallest maximal degree: 3 in 1996
• Largest minimal degree: 2 in 1989, 2016, and 2018
• Largest diameter: 6 in 2009
• Smallest diameter: 2 in 1993, 2000, 2001, 2002, 2007, 2011, and 2016
• Disconnected: 1985, 1991, 1996, 1998, 1999, 2006, and 2008
• Isomorphic seasons: 1991 and 1998
• Hamiltonian cycle: 2016 and 2018
• Triangle-free: none (hence no trees and no bipartite graphs)

Appropriately, both Hamiltonian cycles include Hamilton.

## 1985 season

This was the year of Senna’s first race victory, but the championship went to Prost, who shared maximal vertex degree (4) with Rosberg (Keke Rosberg, of course, not his son Nico Rosberg). This is also one of the few seasons with a disconnected graph. A small connected component, such as Angelis-Boutsen here, likely indicates something weird… in this case, the 1985 San Marino Grand Prix at Imola where Senna ran out of fuel and Prost was disqualified.

## 1986 season

Prost won again, this time with vertex degree 5.

## 1987 season

The four-way battle between Mansell, Piquet, Prost, and Senna fell just short of creating a complete subgraph on four vertices. Their best chance of creating ${K_4}$ was at Detroit, where Senna won and Prost was 3rd. Piquet won the championship.

## 1988 season

The graph is smaller than the previous ones, but is actually larger than one would expect, considering that Senna and Prost combined for 15 wins in 16 races. Berger extended this graph by his win at Monza, in the season otherwise dominated by McLaren. The graph also suggests that Prost should win the championship, and he would have if the champion was determined by the total of all points earned as it is now. But only the best 11 results counted then, and Senna won by that metric.

## 1989 season

Again just an edge short of ${K_4}$ subgraph, but this time it was not a four-way battle at all. Berger only finished 3 races (but in top two every time). Senna and Mansell also had too many retirements to challenge Prost for the championship. This is the first time we see a graph with no vertices of degree 1. But there is no Hamiltonian cycle here.

## 1990 season

The first time we see a degree of vertex 6, and the second time Senna is the champion.

## 1991 season

Another disconnected graph, with Piquet scoring his last career victory in Canada under strange circumstances: Mansell’s car stopped on the last lap when he led by almost a minute and was already waving to the crowd.

If such a mishap also happened at Silverstone, where Mansell, Berger, and Prost finished 1-2-3, we would have ${K_4}$ as a subgraph. Senna won the championship for the last time.

## 1992 season

Sorry about Schumacher’s name being cut off… this was the year of his first race win, at Spa-Francorchamps. Meanwhile, Mansell utterly dominated the championship.

## 1993 season

The first time we get a graph of diameter 2. It suggests Hill was the winner, but in reality he finished third in the championship, with Prost winning for the last time in his career.

## 1994 season

The year of Senna’s death; he does not appear on the graph. Hill has the vertex degree of 5, but Schumacher won the championship by 1 point after their controversial collision at Adelaide.

## 1995 season

That’s pretty close to the wheel graph on six vertices – the only missing edge is Häkkinen-Coulthard. They would score a lot of 1-2 finishes for McLaren in the years to come, but at this time they were not teammates yet. At the center of the incomplete wheel, Schumacher won the championship by a wide margin.

## 1996 season

Another small component, another highly unusual race: wet Monaco Grand Prix, where only three cars made it to the finish and Panis scored the only victory of his career.

Hill won the championship in which no driver had vertex degree greater than 3, the only such season in our record.

## 1997 season

This season holds the record for the number of vertices (12). Two vertices have degree 6 (Villeneuve and Schumacher) but surprisingly, there is no edge between them. Although one of them was on the podium in every race except Italy, they were never on the podium together. Their infamous collision in the season finale at Jerez led to Schumacher being disqualified from the championship.

Villeneuve became the last non-European F1 champion to date.

## 1998 season

The small component is due to Carmageddon on the first lap of very wet Belgian Grand Prix.

This is where my decision to include only driver’s last names backfires: Ralf Schumacher gets to keep his initial. In other news, Williams suddenly faded from the picture and McLaren re-emerged with Häkkinen and Coulthard finishing 1-2 in five races. Häkkinen won the championship.

The seasons 1991 and 1998 is the only pair of isomorphic graphs in this collection. An isomorphism maps Schumacher and Häkkinen to Senna and Mansell.

## 1999 season

The small component is contributed by the partially wet Nürburgring race, where multiple retirements among the leaders left Herbert to score his last Grand Prix victory.

Schumacher’s injury at Silverstone took him out of contention. Still, the second championship of Häkkinen was a lot closer than the first one: he won by 2 points over Irvine.

## 2000 season

Finally, we get a complete subgraph on four vertices: the Ferrari and McLaren drivers. The sole appearance of a driver outside of these two teams was at Brazilian Grand Prix, where Fisichella finished 3rd but was promoted to 2nd after Coulthard’s disqualification. If not for this incident, we would have a regular graph in this collection, a rather unlikely event. Even so, this season set the record for fewest vertices (5). A closely fought championship ended with Schumacher collecting his third title.

## 2001 season

This was not close at all: the driver at the center of this diameter 2 graph won with a lot of room to spare.

## 2002 season

Another season of diameter 2. Schumacher finished every race in top two, except for the Malaysian Grand Prix, narrowly missing an opportunity to create a tree (a star graph). This season ties the fewest edges record (6) which was set in 1998.

## 2003 season

More vertices and larger diameter indicates a more interesting season. Schumacher won again, but by mere 2 points over Räikkönen.

## 2004 season

The final season of Schumacher/Ferrari dominance, in which Schumacher won 13 races and achieved the vertex degree of 6.

## 2005 season

This looks like it was between Alonso and Räikkönen – and it was, with Alonso becoming the youngest F1 champion yet.

## 2006 season

Button’s first career win (wet Hungarian Grand Prix) created the small component.

The large component has diameter 2, with Alonso (the champion) in its center. This is also the last graph in which Schumacher appears.

## 2007 season

As in 2000, Ferrari and McLaren combine to form a complete subgraph on four vertices. But this championship fight was as close as one could imagine, with three drivers finishing within one point: Räikkönen 110, Hamilton 109, Alonso 109. And this was Hamilton’s first season in F1.

## 2008 season

For the first time, we have a small component with more than two vertices. Kovalainen’s only F1 victory came in Hungary, where Glock took second place. More notable was Vettel’s first victory, which came in Monza and made him the youngest driver to win a F1 race [up to that time]. Even more notably, Hamilton won the championship by one point, at the end of the final lap of the final race, and became the youngest F1 champion at that time. Here is the Glock’s view of the action, his car slip-sliding on dry-weather tyres.

On the graph, “Jr.” is Piquet Jr. who took second place in Germany but his brief stint in Formula 1 would be remembered for an entirely different reason.

## 2009 season

The graph of largest diameter (6) captures a strange season after major rule changes. It is so close to being a complex tree, but the 3-cycle was completed at Istanbul, where the polesitter Vettel lost the lead on the first lap and then fell behind his Red Bull teammate Webber as well, finishing just 0.7 seconds behind in the 3rd place. If Vettel was first or second in Turkey, we would have a tree. Button won the championship on the strength of the first half of the season.

## 2010 season

The third time we see a ${K_4}$ subgraph, but the first time that it involves more than two teams: the vertices come from Red Bull (Vettel and Webber), McLaren (Hamilton), and Ferrari (Alonso). Although Vettel’s vertex degree is only 3, trailing Hamilton’s 4 and Alonso’s 5, he became the youngest F1 champion in history, a record he still holds.

## 2011 season

The season tied 2000 for the fewest vertices, with 5. The fewest edges record (6) is tied as well: it was McLaren in 1988 and Ferrari in 2002; this time it is Red Bull’s turn. Vettel won the championship by 122 points but it’s not all in the car; his teammate Webber finished only third.

## 2012 season

With 16 edges, this season beat the previous record set by 1997 season, even though there are fewer vertices here. The two degree-6 vertices led the way in the championship, with Vettel beating Alonso by 3 points. Was this the last great season to watch?

## 2013 season

Vettel over Alonso again, but by 155 points this time. This was the last season of V8 engines, and last season of Red Bull domination. Hamilton appears on the graph only because of his victory in Hungary, after which Vettel won the remaining 9 races. The season opener turned out to be the last race [at the time of writing] won by someone not driving Mercedes, Ferrari, or Red Bull:

## 2014 season

The beginning of a new era: V6 hybrid engine, Mercedes, and Hamilton. Also the last time we see a McLaren driver (Magnussen) on the graph: he appears because of the 2nd place in the dramatic season opener.

In a brief moment of Williams resurgence, Bottas took 2nd place in Britain and Germany, forming a cycle with the Mercedes drivers. If not for him, we would have a tree.

## 2015 season

Another 6-edge graph, another season without much competition. Vettel was the only driver to challenge Mercedes on occasions, thus contributing a cycle to the graph. The entire graph is formed by Mercedes, Ferrari, and Red Bull. Hamilton won the championship again.

## 2016 season

The first time we get a Hamiltonian cycle, for example: Hamilton, Vettel, Rosberg, Räikkönen, Verstappen, Ricciardo, and back to Hamilton. Another 6-vertex graph formed by Mercedes, Ferrari, and Red Bull exclusively. Among them, Mercedes and Red Bull drivers form a complete subgraph. With Ferrari fading to third, neither Vettel nor Räikkönen had enough success to extend ${K_4}$ to ${K_5}$ and thus create the first non-planar season. We would have ${K_5}$ if (a) Räikkönen overtook Verstappen in Austria (he was 0.3s behind), after Hamilton and Rosberg collided on the last lap:

and (b) Räikkönen finished 2nd instead of the 4th in Malaysia, where Hamilton’s engine went up in smoke, costing him the championship.

As it happened, we did not get ${K_5}$ and Hamilton did not get the championship, which went to Rosberg instead. But Verstappen got his first victory at Barcelona and still remains the youngest driver ever to win an F1 race.

## 2017 season

Once again, it is all about Mercedes, Ferrari, and Red Bull, with the Mercedes drivers enjoying higher vertex degree. But this time Ferrari drivers are connected by an edge. The last 1-2 finish of Ferrari to date was in Hungary, arguably their high point of the season.

It was all about Hamilton the rest of the season.

## 2018 season

Second time a Hamiltonian cycle appears, for example: Hamilton, Räikkönen, Verstappen, Vettel, Ricciardo, Bottas, and back to Hamilton. Fourth year in a row that only Mercedes, Ferrari, and Red Bull drivers appear on the graph. Second year in a row that Hamilton wins, and his fifth time overall.