Envy-free two-player mm-cake and three-player two-cake divisions

Cloutier, Nyman, and Su (Mathematical Social Sciences 59 (2005), 26–37) initiated the study of envy-free cake-cutting problems involving several cakes. The classical result in this area is that when there are q players and one cake, an envy-free cake-division requiring only q − 1 cuts exists under weak and natural assumptions. Among other results, Cloutier, Nyman, and Su showed that when there are two players and two or three cakes it is again possible to find envy-free cake-divisions requiring few cuts, under same assumptions. In the present note, we prove that such a result also exists when there are two players and any number of cakes and when there are three players and two cakes. The proof relies on a theorem by Gyárfás linking the matching number and the fractional matching number in m-partite hypergraphs.