# Complete Matching and Maximum Matching

This is the problem I have been struggling with for a while (from Discrete Mathematics and its Applications (Rosen) seventh edition):

Suppose that a new company has five employees: Zamora, Agraharam, Smith, Chou, and Macintyre. Each employee will assume one of six responsibilities: planning, publicity, sales, marketing, development, and industry relations. Each employee is capable of doing one or more of these jobs: Zamora could do planning, sales, marketing, or industry relations; Agraharam could do planning or development; Smith could do publicity, sales, or industry relations; Chou could do planning, sales, or industry relations; and Macintyre could do planning, publicity, sales, or industry relations.

a) Model the capabilities of these employees using a bipartite graph.

b) Find an assignment such that each employee is assigned one responsibility.

c) Is the matching you found in part (b) a complete matching? Is it a maximum matching?

I have been struggling with part C for a while. Would the matching I found in part B be considered a complete matching and/or a maximum matching? Here's what I have so far (ignore part C):

I have found many different definitions but none that adequately answer my question. Thanks for you help!

The matching in (b) is maximum: in a bipartite graph with partitions $X$ and $Y$ the number of edges in a maximum matching is at most $\min(|X|,|Y|)$. Here this last expression works out to 5, and five edges are used.