People are getting together again now that the restrictions are easing. If you and your friends need some help breaking the ice, here is a mathematical party game you can try.
Challenge everyone in your group to shake hands with an odd number of people. If you prefer a little more social distance, you can substitute air high-fives. Let's assume that you and six of your friends are involved.
You all start at an even number after everyone has shaken their hands. Let's say Anna andByron shake each other's hands. They are holding one handshake each.
She shook his hand and gave him one handshake, an odd number. He is back to an even number now that he has two handshakes.
The three of them jump in and start shaking hands.
Your friends figure things out and come up with a winning configuration.
One extra handshake makes a big difference. One of your friends will have an even number of handshakes once you shake hands.
The game is represented as a graph with points and segments between them. The dilemma you face is an example of a simple but profound idea in graph theory, a field of mathematics that studies the properties and features of important representations. Many of the fundamental principles of graph theory were established hundreds of years ago, but scientists still use them to better understand how all sorts of systems are connected. There are new results from active research about the kinds of graphs in the mathematical party game.
In the game, people and handshakes are edges. The number of handshakes each person has is represented by the number of edges that are connected to the other side of the graph.
The graphs that mathematicians study can get very large and complicated, so it helps to have some simple features to look at. The sum of all the degrees of a graph is a feature. Our party game is impossible to win with seven people. Let's see why.
To calculate the degree sum of a graph, you can list all the individual degrees and add them up. There is another way that uses a clever accounting of the edges. The overall degree sum is determined by the number of edges that connect to each other and the number of edges that connect to each other. The same thing can be said about summing all the degrees in a graph. Since the degree sum is twice the number of edges, it always has to be an even number.
Patrick Honner is a high school teacher from Brooklyn, New York.
You can see all the academy columns.
Our party game is doomed by this. It must have seven people and seven odd degrees to win the game. We will get an even number if we add the first two odd degrees. The sum will be odd if you add the next. The sum will be even if you add the next. The sum has to be odd if you add an odd number of numbers. We can make a graph with an odd number of odd degrees because the degree sum of any graph must be even. The party game is not winnable.
If you have an even number of players, you can win the game. We saw this before you shook someone's hand.
We saw it before the first two people shook hands.
It is possible for each of them to shake an odd number of hands if only two people play the game. You can think of this pair as a smaller graph within a larger one with odd degrees.
When constructing a subgraph, you should only consider the edges between the two sets of vertices. You just ignore the edges that are outside your subgraph.
A subgraph splits a graph into two parts, which is something graph theorists like to do when analyzing a graph, and it can help them identify clusters of vertices that are connected in special ways. Additional information about the structure of the graph can be given by knowing that a subgraph is odd. You can always find a path between any two points on a graph if it is connected with an Eulerian circuit.
It isn't always possible to form an odd graph with a set of vertices. It's possible to form an odd subgraph. One boring way to accomplish this is to pick two vertices that connect to each other and ignore the other edges. That is a very small subgraph. Is it possible to find a large subgraph?
It is already known that every graph has a large subgraph. In the 1960s, a Hungarian mathematician showed that every graph can be divided into two subgraphs. One of the two subsets would have to contain at least half the n vertices. Every graph has an even subgraph that is at least half the size of the original.
How big an odd subgraph can be has been an open research question in graph theory for over 60 years. There was an odd subgraph in the failed party game graph.
This is larger than the original. There is another failed attempt at the game with a smaller maximal odd subgraph, one that uses only four of the original vertices.
Some early successes expressed the fraction of the graph used in the search for the largest guaranteed odd subgraph. In the 1990s, the mathematician Yair Caro showed that any graph with n vertices must have an odd subgraph. A graph with 25 vertices must have an odd subgraph that contains at least frac110 of the 25 and a graph with 100 must have an odd subgraph that contains at least frac110 of the 100. Similar results followed, but mathematicians were looking for what Gallai had found: a single fraction that works for odd subgraphs.
In 2021, Asaf Ferber and Michael Krivelevich proved that every graph has an odd subgraph that uses at least frac110,000 of the original graph. This may seem like a low bar to set, since some have speculated that the true fraction is closer to frac27, but finding a single fraction that works allows mathematicians to prove that. One handshake can make a difference.
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This graph has some even- degree vertices and is not an odd one. If we can find an odd subgraph with six vertices, that would be the maximum size. The example is shown below.
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