Nutné podmínky faktorizace

Abstract

This bachelor thesis deals with factorization of complete graphs into the isomorphic trees.This is an overview of the necessary conditions and an overview of trees on 2n vertices, where n=2,3,4,5,6, that do not factorize the complete graphs. Although the factorization is discipline of graph theory, which is solved over 50 years,complete classification of trees is not fully known.

The thesis is divide into two main parts. The first part is an overview of the nessery conditions of factorization, which are taken from articles by several authors \cite{cart_small_diam}, \cite{diam5}, \cite{nonleave}, \cite{decom_fact}, \cite{spanning_tree}, \cite{max_degree}, \cite{brooms} and \cite{diam6}. In the second part is a list of trees, we solve, which necessary condition helps to determinate which trees do factorize or don't factorize complete graph K_{2n}.

The purpose of this thesis is to compare individual necessary conditions, to find more trees, that does not factorize and for which we can not decide even with existing necessary conditions. Based on the examination of these trees we suggest several hypotheses.