Defined as for the third proof of halls theorem

Assignment Help Mathematics

Reference no: EM13926934

1. Let M be a matching in a bipartite graph G. Show that if M is suboptimal, i.e. contains fewer edges than some other matching in G, then G contains an augmenting path with respect to M. Does this fact generalize to matchings in non-bipartite graphs?

Hint: Recall how an augmenting path turns a given matching into a larger one. Can you reverse this process.

2. Derive the marriage theorem from Konig’s theorem.

Hint: If there is no matching of A, then by Konig’s theorem few vertices cover all the edges. How can this assumption help you to find a large subset of A with few neighbors?

3. Let G and H be defined as for the third proof of Halls theorem. Show that dH(b) ≤ 1 for every b ∈ B, and deduce the marriage theorem.

Hint: Show that the marriage condition fails in H for A1 S A2. The proof is almost a mirror image of the third proof, with unions and intersections interchanged.

Reference no: EM13926934

Questions Cloud

Independent t test exercises : The hypothesis being tested is: Women who are working will have a lower level of depression as compared to women who are not working. Using Polit2SetC SPSS dataset, which contains a number of mental health variables, determine if the above hypothesi..

Goals of computer security : Computer security as a whole comprises six goals, but not every goal is important to every application. All Web sites are concerned with access control and availability.

Identify which ethics theory best matches your approach : How does each theory create a different perspective of the situation, and what impact does this have on your decision? Does one theory work better to identify the ‘right' answer?

Characteristics of the criminal incident : The Crime (Describe the characteristics of the criminal incident you found, then describe what we know about this type of crime more generally. . . prevalence, trends, costs, etc.)

Defined as for the third proof of halls theorem : Let M be a matching in a bipartite graph G.Derive the marriage theorem from Konig’s theorem.Let G and H be defined as for the third proof of Halls theorem.

Anxiety usually interferes with performance : Some anxiety is probably healthy; it means you want to do a good job. But too much anxiety usually interferes with performance. Here are some tips you may want to consider as you plan and create your course project.

Significance level decreases the chances of rejecting : In 1996 the mean monthly bill for cell phone users was $45.70 with a standard deviation of $22. In 1997 the mean monthly bill for cell phone use for a sample of 41 phone bills was $42.40. At the 5% significance level, is there sufficient evidence ..

Outside of in-person interaction between individuals : In an experiment with people who use Facebook, we test whether emotional contagion occurs outside of in-person interaction between individuals by reducing the amount of emotional content in the News Feed. When positive expressions were reduced, pe..

Determine the net effect on jenkins pretax profits : The firm's required pretax return on receivables investments is 20 percent. Determine the net effect on Jenkins' pretax profits of offering a 1 percent cash discount.

User Account

All Pages