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Terrell
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there's a proof provided, but i want to know the intuition as to why it is 2^k.
The largest n such that K_n can be expressed as the union of bipartite graph is 2^k where k is the number of bipartite graphsmfb said:I think there is something missing in the question.
there's a proof provided, but i want to know the intuition as to why it is 2^k.
i did some calculations but it still just won't sit well with me. i did some research and found that there's 2^k colors since each bipartite graph have 2 colors and there are k such bipartite graphs. after some thinking, it became obvious that the number of vertices should be less than 2^k because each vertex can appear in more than one bipartite graph thus it can have more than 1 color. lastly, for the equality case, there are calculations that did satisfy that ,but i seem to lack the understanding as to why.QuantumQuest said:##K_n## is a complete graph and the theorem claims that this can be expressed as a union of ##k## bipartite graphs if and only if ##n\leq 2^k##.
So, in order to get a more intuitive view, I'd advise to solve for ##k## and see the minimum number of bipartite graphs sufficient to cover ##K_n##.
You can take some values for ##k## and see where it goes, doing some required representations along the way. The gist of the theorem is that if the inequality gets violated there can't be a coverage of ##K_n## with ##k## bipartite graphs.
K_n refers to the complete graph with n vertices. This means that every pair of vertices in the graph is connected by an edge.
This means that K_n can be broken down into smaller, simpler graphs that, when combined, make up the complete graph with n vertices.
The largest n can be determined by finding the maximum number of edges that can be added to smaller graphs while still maintaining a complete graph with n vertices.
Yes, the smaller graphs must also be complete graphs. This means that every pair of vertices in the smaller graph must be connected by an edge.
Yes, this concept has applications in network design, specifically in determining the minimum number of connections needed for a complete network. It can also be applied in the study of social networks and communication systems.