Consider the below graph for 7. Each node represents a unique partition of 7. I notice that we can reach to any node from the bottom by partitioning only even integer except the bottom node. I have tried it for other numbers. For instance, number 5,6,8 and found that it works. Will this works for any number or not I do not know? mathematically I am unable to prove it. Could you please help me regarding this. 
2026-03-30 17:15:33.1774890933
Proof for this partition graph
137 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
1
Consider any partition $P$ that has at least $3$ pieces. At least two of those pieces have the same parity. Pick two pieces with the same parity and merge them, i.e., replace them by their sum. Since the sum is even, the new partition $P'$ that you produced is joined to $P$ by an edge in your "split only even integers" graph. Continuing in this way, you can produce a path in your graph, heading from $P$ downward toward the root, until you get down to a partition with only two pieces. Then, you're just one step from the root, and your graph has an edge to the root even if the root is odd.