Matching in Graphs

Question

How can I find a graph which has a maximum cardinality matching of size n/4? or say n/3? Here, n denotes the number of vertices in the graph. Is it possible for a connected graph?

Answer 1

The simplest example is a complete bipartite graph K_m,n . To adjust matching size to be of some proportion ( |V|/k ), n should be (2*k-1)*m , since then |V| = n + m = 2*k*m |V| = n + m = 2*k*m , and matching size is 2*m .

Answer 2

You can construct such graphs based on the Tutte Berge formula . Using this you can construct a graph with maximum matching size n/4, as follows. Let V be the set of vertices in the graph and let U be any subset of vertices. Find a number k such that k-|U| >= |V|/2. Construct odd-sized components C1, C2,...,Ck (that are disjoint with each other) from VU and add any set of edges from each Ci to the vertices of U.

Note that C1,...,Ck are supposed to be the odd sized components that we get upon deleting U. Other vertices in {U, C1, ... ,Ck} can be connected in any way subject to the above constraint.