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a) Given a set of positive integers, its common divisor graph (CD-graph) is the graph whose vertices are the integers, two of which are joined by an edge if (and only if) they have a common divisor greater than 1. Find a set of four positive integers whose CD-graph is the graph on the left below, and such that if the Collatz recipe (multiply by 3, and add 1 if odd; divide by 2 if even) is applied to each of the set's elements, the CD-graph of the new set of numbers is the graph on the right.

b) Are any two graphs on 4 vertices convertible one into the other in a similar way?

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a)

My choice is: 3,9,110=2*55, 154=2*77. Here, 3 and 9 are connected, as are 110 and 154 as both have 11 as prime factor. These numbers are mapped to 10, 28, 77, 55. In this chain of numbers we have the connections 10-28-77-55-10, so we get the graph on the right.

b)

No, the graph with four vertices and no edges will be mapped onto a graph with at least three edges: at most one of the numbers of the first set can be even, so at least three of them are odd. After applying the Collatz map at least three numbers are even, and thus connected.

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  • $\begingroup$ It would be interesting to know how many of the 121 pairs of graphs on 4 vertices are convertible. Almost all, surely. $\endgroup$ – Bernardo Recamán Santos Feb 21 at 21:00

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