Topološki i diskretni pristup problemu podjele ogrlice ; Topological and discrete approach to the necklace splitting problem

Cilj ovog rada je riješiti neke varijante problema podjele ogrlice koristeći različite strategije. U tu svrhu smo prvo formalizirali problem, odnosno uveli definiciju ogrlice i pravedne podjele ogrlice i iskazali teorem o egzistenciji pravedne podjele. Prvi pristup koji smo koristili za rješavanje problema je topološki. Definirali smo homotopiju, fundamentalnu grupu i odredili fundamentalnu grupu kružnice s ciljem dokazivanja Borsuk-Ulam teorema. Konačno, iskoristili smo Borsuk-Ulam teorem za dokazivanje egzistencije pravedne podjele ogrlice. Nadalje, promotrili smo neke posebne slučajeve problema i dokazali ih direktno i u tom procesu iskoristili određene rezultate iz teorije grafova. Naposlijetku, definirali smo kubične komplekse i promatrali kubična preslikavanja. Konstruktivno smo dokazali Ky-Fanov teorem i primjenili konstrukciju iz dokaza na rješavanje problema pravedne podjele ogrlice. ; Goal of this paper is to solve some variants of necklace-splitting problem using different strategies. In this context, we firstly formalized the problem, i.e. introduced the definition of the necklace and fair division of the necklace and stated the theorem on the existence of fair division. The first approach we used to solve the problem was topological. We defined a homotopy, a fundamental group and determined a fundamental group of a circle with the aim of proving the Borsuk-Ulam theorem. Finally, we used the Borsuk-Ulam theorem to prove the existence of a fair division of the necklace. Furthermore, we looked at some special cases of the problem and proved them directly and in the process used certain results from graph theory. Finally, we defined cubical complexes and observed cubical mappings. We constructively proved the Ky-Fan theorem and applied the construction from the proof to solve the problem of fair division of the necklace.