Odd Party Conjecture: Can You Prove or Disprove?

[FaustoMorales]

Conjecture: Consider any group with an even number of people where each member is connected to any other through some chain of people. Then the original group can be split into groups where each member knows an odd number of people directly.

Note: If the party is such that each member knows an odd number of people to begin with, then the null splitting (no splitting at all) does the job.

Can anyone prove this (or find a counter-example)? Good luck!

 

[davee123]

I think there needs to be some clarification--

Reworded:

Given: A group exists with a non-zero, even number of people. Each person in the group directly knows at least one other person within the group. Further, each person "knows" each other person in the group, either directly or indirectly (via people they know directly).

Conjecture:The group can be split into 1 or more subgroups wherein each member of a subgroup knows an odd number of people directly within that subgroup.DaveE

 

[FaustoMorales]

Thanks DaveE for your reply.

Your clear rephrasing is quite helpful.