I am trying to solve the following problem about maxflow mincut it seems like my conclusion is incorrect and I am wonder where.

There is no graph just following question.

An edge e can be (x) always full, (y) sometimes full, (z) never full; it can be (x’) always crossing, (y’) sometimes crossing, (z’) never crossing. So there are nine possible combinations: (xx’) always full and always crossing, (xy’) always full and sometimes crossing, and so on. Or are there? Maybe some possibilities are impossible. Let’s draw a table:

Possible answers —based on my conclusions

**always full and always crossing.** (true based on the image above)
**always full and sometimes crossing.** (true based on the image above and the fact that we may have several crossing edges one of them may be full, all them if all edges in the min cut is 1 and one of the crossing edges is 2)
**always full and never crossing.** (false seems like there always must be a possibility of one of the ways to use edge with less capacity)
**sometimes full and always crossing.** (true if we have all edges in the mincut that are 1 and crossing edge is two)
**sometimes full and sometimes crossing.**( false since if we have many crossing edges that are 1 and one of them is 2 means that we might want to use the edge 2to flow more)
**sometimes full and never crossing.** (true easy to see and edge that sometime full but never crossing in a graph)
**never full and always crossing.** (true if all edges in a mincut is 1 and crossing edge is two)
**never full and sometimes crossing.** (true if we have many 2 crossing edges)
**never full and never crossin**g. (true if we have a graph where one edge is 2 in min cut that never crossing)

Have asked this question on /math.stackexchange.com with no luck.

wording used.

The edge is full means in in any way we pump the flow the edge is always going to be full, basically if we have and C—(1)—A——(1)—-B and there is no other way to get to B other then going thru a then A is always full. Somewhat full if we can have C—(1)—A——(1)—-B and if there is another way to get to b by doing C—(1)—E—(1)—-B then edge is only sometime full. never full if we have C—(1)—E—(2)—-B then most we can flow is 1 but b takes 2 then it never full. The crossing part if we it can be the crossing edge in the min-cut from the cut to the sink. basically it and edge that connects the cut with the rest of flow.

Thanks.