Great explanation with the visuals! The flowchart at 9:32 helps in summarizing the proof pretty well and it enables one to list all cases without missing any.
At 4:50 in the video, you mentioned "There is no alternating path that ends in B3," but why isn't the edge (A5, B3) considered an alternating path that ends in B3? This is because you also stated at 7:00 in the video that the edge (A2, B2) is an alternating path ending in B2.
I love this video... did you consider starting with the problem and finding all the things needed as a motivation? (It might have made the video longer and perhaps it is not possible with that problem because there are many concepts to be introduced but it is an interesting thought)... In any case, what i like about the video the most is that it exemplifies how to read a proof... Many proofs on Diestel are like that, in order to read the proof, you have to sit down and draw things and try to figure out why Diestel wrote things that way (most likely due the limitation of our linear language to express things that are better suited for drawings or animations). I haven't read this proof in a while but i think it only takes like a page of Diestel but I remember that when I read it, it took me like half an hour to finally understand it... Great job! I hope this video find its audience and go viral!
🎯 Key Takeaways for quick navigation: 00:00 A *bipartite graph has vertices divided into two groups, A and B, with no edges within the same group. It is said to be bipartite.* 00:52 Kőnig's *theorem states that, for any bipartite graph, the size of its maximum matching (matched vertices) is equal to the size of its minimum vertex cover (covering all edges).* 03:11 Understanding *alternating paths and augmenting paths is crucial for the proof of Kőnig's theorem.* 04:09 The *proof involves constructing a set U from the maximum matching, showing that U covers all edges, and establishing its equality in size with the maximum matching.* 09:50 Kőnig's *theorem has practical applications, such as determining the maximum number of students that can be placed in a classroom without cheating by finding the maximum independent set in a graph.* Made with HARPA AI
At 1:24 I think you've made an error in your definition of a minimum vertex cover in saying that it is "a vertex cover with the smaller possible number of edges, i.e. there is no other vertex cover with less edges." There is no such thing as a vertex cover with more or less edges, the definition of a vertex cover implies that the vertices of a vertex cover are incident to |E(G)| edges. If there are less edges than this, it is by definition not a vertex cover of G.
Great explanation with the visuals! The flowchart at 9:32 helps in summarizing the proof pretty well and it enables one to list all cases without missing any.
thanks for saving my competitive programming journey :)
Great explanation! Thanks! Good to see you be more active again :)
Very good video, great graphics as well
At 4:50 in the video, you mentioned "There is no alternating path that ends in B3," but why isn't the edge (A5, B3) considered an alternating path that ends in B3?
This is because you also stated at 7:00 in the video that the edge (A2, B2) is an alternating path ending in B2.
I was wondering the same thing but found out that an alternating path starts with an unmatched vertex after looking around a bit.
Nice explanation and Visuals help a lot.
Thanks a lot
I love this video... did you consider starting with the problem and finding all the things needed as a motivation? (It might have made the video longer and perhaps it is not possible with that problem because there are many concepts to be introduced but it is an interesting thought)... In any case, what i like about the video the most is that it exemplifies how to read a proof... Many proofs on Diestel are like that, in order to read the proof, you have to sit down and draw things and try to figure out why Diestel wrote things that way (most likely due the limitation of our linear language to express things that are better suited for drawings or animations). I haven't read this proof in a while but i think it only takes like a page of Diestel but I remember that when I read it, it took me like half an hour to finally understand it... Great job! I hope this video find its audience and go viral!
Well explained thank u
You are welcome
Thanks for the video! In 4:56 you say, that there is no alternating path ending in B4, but there is one between a4 and b4, isn't that so?
🎯 Key Takeaways for quick navigation:
00:00 A *bipartite graph has vertices divided into two groups, A and B, with no edges within the same group. It is said to be bipartite.*
00:52 Kőnig's *theorem states that, for any bipartite graph, the size of its maximum matching (matched vertices) is equal to the size of its minimum vertex cover (covering all edges).*
03:11 Understanding *alternating paths and augmenting paths is crucial for the proof of Kőnig's theorem.*
04:09 The *proof involves constructing a set U from the maximum matching, showing that U covers all edges, and establishing its equality in size with the maximum matching.*
09:50 Kőnig's *theorem has practical applications, such as determining the maximum number of students that can be placed in a classroom without cheating by finding the maximum independent set in a graph.*
Made with HARPA AI
You have changed the definition of augmenting path and alternating path here.
Can you clarify more?
I think you need to say that the first vertex needs to be unmatched in your definition of alternating path.
Thanks for mentioning it
At 1:24 I think you've made an error in your definition of a minimum vertex cover in saying that it is "a vertex cover with the smaller possible number of edges, i.e. there is no other vertex cover with less edges." There is no such thing as a vertex cover with more or less edges, the definition of a vertex cover implies that the vertices of a vertex cover are incident to |E(G)| edges. If there are less edges than this, it is by definition not a vertex cover of G.