a curiosity about cards, why this is true?

Question

We split the standard 52-card deck between 13 people, each of them gets 4 cards.

In any case it is possible to take away one card from each person to fulfil the following condition: the 13 cards taken away from them are of 13 different values.

I found this in a math magazine as just a curiosity. How this can be proven? (I have completely no idea)

Consider a graph with $26$ vertices, consisting of the $13$ players and the $13$ card ranks, where each player is joined by an edge to the four ranks they are holding (if a player holds three nines, they are joined to the nine vertex via three edges). This is a $4$-regular bipartite graph, so it has a perfect matching. — Mike Earnest, Sep 18 '21 at 19:31

score 2 · Answer 1 · answered Sep 18 '21 at 21:50

Consider a graph with $26$ vertices, consisting of the $13$ players and the $13$ card ranks, where each player is joined by an edge to the four ranks they are holding. This is technically a "multi-graph:" if a player holds three nines, they are joined to the nine vertex via three edges.

Each player holds four ranks, and each rank appears in four hands (possibly with repeats), so this is a $4$-regular bipartite graph. Using the answers to this MSE question, we can show that this graph has a perfect matching, which corresponds to how each player can choose a rank with no two players choosing the same.

a curiosity about cards, why this is true?

1 Answers1