TITLE: P-graph of partition elements of 100 under common divisibility relation
QUESTION [9 upvotes]: Given a multiset of positive integers, its P-graph is the loopless graph whose vertex set consists of those integers, any two of which are joined by an edge if they have a common divisor greater than 1, that is, they are not relatively prime.
The P-graph of a certain mystery partition of 100 has 10 vertices and 42 edges.  I've been told that no other partition of 100 has exactly the same P-graph.  What is this mystery partition of 100?

REPLY [4 votes]: I assumed no repetitions in the partition and no self-loops in the graph and found none that satisfied the requirements.
If repetitions are allowed in the partition there are 2977866 possible partitions of length 10 to explore.  Regardless, it then becomes unclear whether two vertexes with the same integer should be linked.
Regardless, here is a graph with repetition in the partition having $\{15,12,10,10,10,10,10,10,8,5\}$ having 42 edges:

The 10a, 10b, etc. distinguish nodes that correspond to the repeated element 10.
Here is the graph found by Smylic:

Bernardo shows that they are not isomorphic.