Given a necklace with $d$ types of beads and $p$ people it is well known we can fairly divide the necklace with at most $d(p-1)$ cuts. A fair division means that each person is given the same number of beads of type $i$ where $i \in \{1, \dots, d\}$. We also assume that the number of beads of type $i$ is divisible by $p$ for all $i \in \{1, \dots, d\}$.

I am curious if it is known which necklaces require $d(p-1)$ cuts. I cannot seem to find it in any of the literature but maybe one of you is aware of this result. Clearly a necklace which groups all of the beads of a type together will require the maximum number of cuts. You can also vary this idea and repeat a necklace of this type a specific number of times based on $p$. Is this it?

Edit: The proof of $d(p-1)$ cuts being optimal for $p$ people and $d$ beads is located here: http://ac.els-cdn.com/0001870887900557/1-s2.0-0001870887900557-main.pdf?_tid=ab3e4b8a-6bf5-11e7-8a41-00000aab0f27&acdnat=1500409034_155853417bd79ecefa7dbb07d38e83e9

Given an unclasped necklace with $d$ types of beads and $p$ people it is well known we can fairly divide the necklace with at most $d(p-1)$ cuts. A fair division means that each person is given the same number of beads of type $i$ where $i \in \{1, \dots, d\}$. We also assume that the number of beads of type $i$ is divisible by $p$ for all $i \in \{1, \dots, d\}$.

I am curious if it is known which necklaces require $d(p-1)$ cuts. I cannot seem to find it in any of the literature but maybe one of you is aware of this result. Clearly a necklace which groups all of the beads of a type together will require the maximum number of cuts. You can also vary this idea and repeat a necklace of this type a specific number of times based on $p$. Is this it?

Edit: The proof of $d(p-1)$ cuts being optimal for $p$ people and $d$ beads is located here: http://ac.els-cdn.com/0001870887900557/1-s2.0-0001870887900557-main.pdf?_tid=ab3e4b8a-6bf5-11e7-8a41-00000aab0f27&acdnat=1500409034_155853417bd79ecefa7dbb07d38e83e9

Given a necklace with $d$ types of beads and $p$ people it is well known we can fairly divide the necklace with at most $d(p-1)$ cuts. I am curious if it is known which necklaces require $d(p-1)$ cuts. I cannot seem to find it in any of the literature but maybe one of you is aware of this result. Clearly a necklace which groups all of the beads of a type together will require the maximum number of cuts. You can also vary this idea and repeat a necklace of this type a specific number of times based on $p$. Is this it?

Given a necklace with $d$ types of beads and $p$ people it is well known we can fairly divide the necklace with at most $d(p-1)$ cuts. A fair division means that each person is given the same number of beads of type $i$ where $i \in \{1, \dots, d\}$. We also assume that the number of beads of type $i$ is divisible by $p$ for all $i \in \{1, \dots, d\}$.

I am curious if it is known which necklaces require $d(p-1)$ cuts. I cannot seem to find it in any of the literature but maybe one of you is aware of this result. Clearly a necklace which groups all of the beads of a type together will require the maximum number of cuts. You can also vary this idea and repeat a necklace of this type a specific number of times based on $p$. Is this it?

Edit: The proof of $d(p-1)$ cuts being optimal for $p$ people and $d$ beads is located here: http://ac.els-cdn.com/0001870887900557/1-s2.0-0001870887900557-main.pdf?_tid=ab3e4b8a-6bf5-11e7-8a41-00000aab0f27&acdnat=1500409034_155853417bd79ecefa7dbb07d38e83e9