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Petersen coloring and variants
Bílková, Hana ; Šámal, Robert (advisor) ; Dvořák, Zdeněk (referee)
The Petersen coloring of 3-regular graph G is equivalent to the normal coloring by five colors. The normal coloring is a good coloring of edges such that every edge and its four neighbours have together three or five different colors. Jaeger conjectures that every bridgeless 3-regular graph has a Petersen coloring. If the conjecture were true, it would imply other interesting statements about 3-regular graphs. In this text we investigate normal coloring by more than five colors. Jaeger theorem about nowhere-zero Z2 3 -flow implies that every bridgeless graph has normal coloring by seven colors. Independently on the Jaeger theorem, we prove the existence of normal coloring by nine colors for graphs with a bridge, a cut of size two or with a triangle. The idea of our proof comes from Andersen's proof of existence of strong coloring by ten colors for 3-regular graphs. Finally, we sketch the idea of the proof for other classes of 3-regular graphs. 1
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Value of education - Cost Benefit Analysis of Methodic "Good Start"
Bílková, Hana ; Sieber, Martina (advisor) ; Vlček, Josef (referee)
The thesis is about evaluating of an investment using Cost Benefit Analysis. The investment that is evaluated in this thesis educational intervention in pre-school education. The evaluation is ex ante. The thesis is divided into three thematic units. In the first part, it deals with the theoretical aspects of CBA, i.e., the description of the analysis and its basic principles. The second part is devoted to methodology, first focusing on the concept of shadow prices, then specific methods for evaluating the investment as such. In the third part the educational methodology itself is evaluated. Cost and benefits are determined where the benefits are determined using a shadow price determined based on the market price method. Based on the result of CBA, it can be said that the value of the intervention is positive. Since there is no normal use od paid educational establishments in our territory, a second value of education is set, which corresponds more to the real social value. With such an education values, the value of the intervention is negative.
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Tax Kobra assessment
Bílková, Hana ; Sieber, Martina (advisor) ; Vlček, Josef (referee)
The theme of my bachelor thesis is the evaluation of an investment. More specifically, the investment that I am going to evaluate is a state project of the Czech Republic called Tax Cobra, which is a little bit different in reality than in my thesis. The bachelor thesis is divided into two parts: the first is theoretical, whereas the second is practical. In the first part, I explain what the Tax Cobra is, how and why this project was made, and who the members of it are. I also explain how the investment is evaluated. I summarize the individual methods of investment valuation and other necessary indicators that are important to take into account, such as the cost per capita lor risk. There is also a description of the financial plan. In the practical part, I valuate the project by using the method described in the theoretical part. There is a substantial description of the financial plan, which is crucial for the evaluation. Following this description, I assert the methods of investment evaluation. Ultimately, I would recommend this investment based on these evaluations.
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Variants of Petersen coloring for some graph classes
Bílková, Hana ; Šámal, Robert (advisor) ; Rollová, Edita (referee)
Normal coloring - an equivalent version of Petersen coloring - is a special proper 5-edge-coloring of cubic graphs. Every edge in a normally colored graph is normal, i.e. it uses together with its four neighbours either only three colors or all five colors. Jaeger conjectured that every bridgeless cubic graph has a normal coloring. This conjecture, if true, imply for example Cycle double cover conjecture. Here we solve a weakened version of Jaeger's problem. We are looking for a proper 5-edge-coloring such that at least a part of the edges is normal. We show a coloring of generalized prisms with two thirds of the edges normal and a coloring of graphs without short cycles with almost half of the edges normal. Then we propose a new approach to normal coloring - chains. We use chains to prove that there cannot be only one single mistake in an almost normally colored graph. We also prove some statements about cuts in a normally colored graph which also follow from nowhere-zero Petersen flow. Finally, we examine a four-cycle in a normally colored graph. 1
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Petersen coloring and variants
Bílková, Hana ; Šámal, Robert (advisor) ; Dvořák, Zdeněk (referee)
The Petersen coloring of 3-regular graph G is equivalent to the normal coloring by five colors. The normal coloring is a good coloring of edges such that every edge and its four neighbours have together three or five different colors. Jaeger conjectures that every bridgeless 3-regular graph has a Petersen coloring. If the conjecture were true, it would imply other interesting statements about 3-regular graphs. In this text we investigate normal coloring by more than five colors. Jaeger theorem about nowhere-zero Z2 3 -flow implies that every bridgeless graph has normal coloring by seven colors. Independently on the Jaeger theorem, we prove the existence of normal coloring by nine colors for graphs with a bridge, a cut of size two or with a triangle. The idea of our proof comes from Andersen's proof of existence of strong coloring by ten colors for 3-regular graphs. Finally, we sketch the idea of the proof for other classes of 3-regular graphs. 1
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