Pls Help me :)

[limpid moon]

Let ( k \in \mathbb{N} ) be a constant. Show that for a given undirected graph ( G ), it is possible to determine a matching ( M ) in polynomial time such that there is no ( M )-augmenting path in ( G ) with fewer than ( k ) edges. Determine the maximum factor by which such a matching can be smaller than a cardinality-maximal matching. Prove that the bound you establish is optimal.

[restive lily]

1. Do not ping the Moderators, unless someone is breaking the rules.
2. Do not ping the Helper Moderators, unless there is a conflict between helpers.
3. Do not ping other members randomly for help.
4. Ask your question and show the work you've done so far. If you've posted a screenshot of a question, specify which part you need help with.
5. Wait patiently for a helper to come along.
6. If the Helper has answered your question, remember to thank them with the Mathematics Ranks bot and close the thread with:

+close
Feel free to nominate the person for helper of the week in #helper-nominations
If you're happy with the help you got here, and the server overall, you can contribute financially as well:

[covert olive]

tautau

[vital mural]

@limpid moon

:HelpIcon:| Help Reminder

Hello eliashkp, this is a friendly reminder that your help request has been inactive for more than 24 hours. If you no longer need assistance, please consider closing the thread using the +close command. This thread will be automatically closed in 3 days if it remains inactive.