Math 416, Homework 3
Due: 03/20
Assume that all graphs are simple.
Mat 416 students: Problems 1-6,
Mat 513 students: Problems 4-9.
1. Let M be a maximal matching and M ∗ a maximum matching. Show
that |M | ≥ |M ∗ |/2.
2. Let G be a k-regular bipartite graph Show that G has an l-factor for
every l ≤ k.
3. Let G be a bipartite graph with bipartition X, Y . Show that if for
every non-empty set S ⊂ X, |N (S)| > |S|, then every edge of G is in
a matching which saturates X.
4. Let G be a bipartite graph with bipartition X, Y , |X| = |Y | such
that δ(G) ≥ k and for every X 0 ⊂ X, Y 0 ⊂ Y , if |X 0 |, |Y 0 | ≥ k, then
|E(X 0 , Y 0 )| > 0. Show that G has a perfect matching.
5. For every k ≥ 2, construct a k-regular graph without a perfect matching.
6. Let G be a k-regular graph of even order which stays connected when
any k − 2 edges are deleted. Show that G has a perfect matching.
7. Let M be a matching and let u be an M -free vertex. Suppose further
that there are no M -augmenting paths in G that start at u. Show that
there is a maximum matching M ∗ such that u is M ∗ -free.
8. Let G be a 3-regular graph with no cut-edges. Show that the edges of
G can be partitioned into copies of P4 (a path on four vertices).
9. Let G be a connected 2d-regular (d ≥ 1) graph with an even number
of edges. Show that G has a d-factor.
[Bonus/Research] Recall that if G is a graph of even order n with minimum
degree n/2 then G has a Ham cycle and so a perfect matching. A 3-uniform
hyper-graph is a pair (V, E) where E (set of edges) is a set of 3-element
subsets of V . A matching in a hyper-graph is a set of edges which are
pair-wise disjoint. Let n ∈ 3Z be large enough and let H be a 3-uniform
hyper-graph on n vertices.
• For two vertices u, v ∈ V let N (u, v) be the set of hyper-edges in H
containing {u, v}. Show that if minu,v |N (u, v)| ≥ n/2 then H has a
perfect matching.
• Show that the above degree condition is tight.
• For a vertex u ∈ V , let N (u) be the
set of hyper-edges in H containing
5 n
u. Show that if minu |N (u)| ≥ 9 2 , then H has a perfect matching.
• Show that he above condition is tight.