The following problem is taken from Graph Theory by Adrian Bondy, U.S.R. Murty, crediting H. Bass.
m identical pizzas are to be shared equally amongst n students. The pizzas are not necessarily divided into equal parts, and the slices of a given pizza may vary in size.
a. Show how this goal can be achieved by dividing the pizzas into a total of $m + n − d$ pieces, where $d=\gcd(m,n)$.
b. By considering a suitable bipartite graph, show that no division into a smaller number of pieces will achieve the same objective.
Try the example of $m=6$ pizzas shared by $n=8$ students. We can divide each pizza into two slices of sizes $3/4$ and $1/4$, as follows:
Where $6$ students will get one slice of $3/4$, and the other two will eat $3$ slices of $1/4$. I am trying to formulate the idea of slicing each pizza into $n$ slices and "rejoining" some of them into a bigger slice, but I couldn't find the general algorithm for that.