The complete graph K_n can be expressed as the union of k bipartite graphs if and only if n ≤ 2^k, where k represents the number of bipartite graphs. The discussion emphasizes the importance of induction in proving this theorem, starting from k = 1 and proceeding to k = m. Participants are encouraged to use vertex coloring and to divide the vertices of K_n into two disjoint sets for the inductive step when n ≤ 2^(m+1).
PREREQUISITESMathematicians, computer scientists, and students studying graph theory or combinatorial mathematics who seek to understand the relationship between complete graphs and bipartite graphs.
Terrell said:I would simply like to know how to get 2^k
i mean why it is like this.QuantumQuest said:Do you mean why is it like this or how to pick them as a step in the induction needed?
Terrell said:i mean why it is like this
Terrell said: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 graphs
will take it from there. thanks! how do you know it was me? lol... will try proving it later on as i am working on something else. will make sure to get back to this thread.QuantumQuest said:As I remember from another thread you asked for the intuition of
and you got some intuition using coloring. So now for the theorem you have to apply induction on ##k## (##n \geq 2##) in order to prove it.
Start from ##k = 1##.Can you see the truth of the statement?
Proceed to ##k = m## for the inductive step. Hint: For ##n \leq 2^{m+1}## try to divide the vertices of ##K_n## in a helpful manner into two disjoint sets. Can you proceed further?