# 2.11 Stable Matching

## Match or No Match

1. A Perfect Matching

Find a perfect matching the bipartite simple graph G whose vertices and edges are given by the following sets:
V := {a,b,c,d} ∪ {1,2,3,4}
E := {{a,1}, {a,3}, {b,2}, {c,3}, {c,4}, {d,1}, {d,2}}

Input the edges of the matching in increasing alphabetic order, i.e. for each edge, you should write the letter in the pair first. Also list the edges separated by spaces. For example, if you want to answer {d, 1} and {c, 3}, type

(c 3) (d 1)
Exercise 1
In this graph, the perfect matching happens to be unique.
2. No Perfect Matching

The bipartite simple graph G whose vertices and edges are given by the folowing sets:

V := {a,b,c,d} ∪ {1,2,3,4}
E := {{a,3}, {a,4}, {b,1}, {b,2}, {c,1}, {c,2}, {d,1}, {d,2}}
does not have a perfect matching.

Which of the following properties of G make a perfect matching impossible?

1. The set {1,3,4} has only 2 neighbors.
2. The vertices a,b,c,d, on the "left" side, all have degree 2, but none of the vertices 1,2,3,4, on the "right" side, has degree 2.
3. The set {b,c,d} has only 2 neighbors.
4. G has 8 edges.
5. The set {3,4} has only 1 neighbor.
6. Vertex 1 has degree 3, but each of its neighbors only has degree 2.
7. The set {a,b} has 4 neighbors.

Give your answer as a sequence of numbers separated by some spaces
(e.g., "6 9"). Don't use commas or parentheses.

Exercise 2
If we let the letters be the "girls" and the numbers be the "boys", then property 3. is a bottleneck that prevents a perfect match. If we let the letters be the "boys" and the numbers be the "girls", then property 5. is another bottleneck in G.

Property 1. also describes a bottleneck, but it's not true :-)