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given my continuous struggle with proofs on graph theory, I come with another problem I do not know how to approach.

Given a graph G = (V, E) such that for any two non-neighboring vertices u, v ∈ V , d(u) + d(v) ≥ n. (n being the number of vertices) 1. Assuming G is not a clique, what is the diameter of such a graph? 2. Prove that in G there always exists a Hamiltonian cycle.

Thank you to any kind soul willing to help

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  • $\begingroup$ You can always make more of an effort (that is, any effort at all) even if you don't know what to do. For example: for (1), have you tried drawing small examples of such graphs, and what is their diameter? for (2), what techniques do you know to find a Hamiltonian cycle? $\endgroup$ – Misha Lavrov Jan 25 at 15:18
  • $\begingroup$ I have indeed and it seems to me that in every case (that is for 4,5,6 vertices) the diameter is always 2. But I wouldn't know how to justify it. I assume induction can be used to generalise it to more vertices but its something I still struggle with $\endgroup$ – totoro01 Jan 25 at 15:20
  • $\begingroup$ It is better if you add your attempts to the question to begin with. $\endgroup$ – Misha Lavrov Jan 25 at 16:43
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To see that the diameter of this graph is always $2$, just take any two vertices $u$ and $v$ and look at the length $1$ and length $2$ paths between them.

If there is an edge $uv$, they are at distance $1$ and everything is good.

If there is no edge $uv$, then the condition $\deg(u) + \deg(v) \ge n$ applies. There are $n-2$ different possible length $2$ paths from $u$ to $v$, using the edges $ux$ and $xv$ for some third vertex $x$. Now apply the pigeonhole principle:

  • There are at least $n$ edges incident to $u$ or to $v$.
  • There are $n-2$ vertices $x \ne u,v$ in the graph.
  • So at least $2$ of the $n$ edges must be incident to the same vertex $x$.

This gives us a path $ux, xv$ of length $2$.

You see that this proof actually works if we have $\deg(u) + \deg(v) \ge n-1$. We need $\deg(u) + \deg(v) \ge n$ for non-adjacent $u$ and $v$ for the second part of your question, which is just the statement of Ore's theorem.

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