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So we need to show that there is a graph of 17 vertices which contains neither.

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Seventeen vertices is a lot, and even if we successfully drew such a graph, we would not be able to just look at it to verify it—there would simply be too many edges for us to make sense of. Aniqng of drawing a graph, then, we can construct one by defining it in a Discreet fun needed feb 27 Anqing 2 way. Consider a graph of vertices, which are labelled with the integersplaced in a circle.

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We can now decide which vertices are connected by referring to their labels. For example, consider a graph of 7 vertices.

Since we drew them in a circle, this means that 1 shares an edge with 4, 2 shares an edge with 5, etc. Since we have defined our circle such that vertex 7 appears right before vertex 1, we would also needer that 7 shares an edge with 3.

This is analogous to modular arithmetic. Here is what our graph looks like with the blue edges:.

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All other edges should be coloured red. How can we define this without drawing it? Well, two vertices have a blue edge if they are 3 away from one another.

Continuing in this way, we get that two vertices share a red edge if and only if they are 1 or 2 away from each other. Can we do the same for distances of 4, 5, 6, and 7?

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What about 4? Start at any vertex, and go 4 to the right. Now starting at that same vertex, go 3 to the left. Any vertex that is 4 or more away from another vertex is actually a smaller distance away from that same vertex going in the other direction.

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So a distance of 4 is equivalent to a distance of 3, in the way we defined this graph. More generally, a distance of is the same as a distance of. So this gives us a complete graph.

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Now how can we check for mutual friends or strangers? Product details Paperback: Posts and Telecom Press; 2nd edition January 1, Language: Chinese ISBN Be the first to review this item Would you like to tell us about a lower price?

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