Povezavno 3-obarvljivi grafi : diplomsko delo

Šere, Nina (author), Šparl, Primož (mentor)

V diplomskem delu se ukvarjamo s kromatičnim indeksom kubičnih grafov, kjer se omejimo na večji del dobro znane družine takšnih grafov, znanih pod imenom posplošeni Petersenovi grafi. Graf Γ je k-povezavno obarvljiv, če se da njegove povezave obarvati s k barvami tako, da so incidenčne povezave obarvane z različnimi barvami. Najmanjše tako število k imenujemo kromatični indeks grafa in ga označimo χ'(Γ). Ker so posplošeni Petersenovi grafi kubični, ima vsak izmed njih po dobro znanem Vizingovem izreku kromatični indeks bodisi enak 3 bodisi 4. Rezultati tega diplomskega dela predstavljajo pomemben del dokaza, da je znameniti Petersenov graf edini posplošeni Petersenov graf, ki ni povezavno 3-obarvljiv. Z drugimi besedami, Petersenov graf GP(5,2) je edini posplošeni Petersenov graf s kromatičnim indeksom 4.

• Language: Slovenian
• Keywords: barvanje povezav, kromatični indeks, kubični graf, posplošeni Petersenov graf
• Work type: Bachelor thesis/paper
• Typology: 2.11 - Undergraduate Thesis
• Organization: PEF - Faculty of Education
• Publisher: [N. Šere]
• Year: 2017
• Number of pages: V, 29 str.
• PID: 20.500.12556/RUL-95197
• UDC: 519.17(043.2)
• COBISS.SI-ID: 11704905
• Publication date in RUL: 19.09.2017

ŠERE, Nina, 2017, Povezavno 3-obarvljivi grafi : diplomsko delo [online]. Bachelor’s thesis. N. Šere. [Accessed 27 April 2025]. Retrieved from: https://repozitorij.uni-lj.si/IzpisGradiva.php?lang=eng&id=95197

Secondary language

• Language: English
• Title: 3–edge colorable graphs

Abstract

In this BSc thesis we deal with chromatic index of cubic graphs, where we mainly focus on a significant part of the family of graphs, named generalized Petersen graphs. A graph Γ is said to be k-edge-colorable, if we can color its edges with k colors, so that incident edges are colored with different colors. The smallest such number k is called the chromatic index and it is denoted by χ'(Γ). Due to the fact that generalized Petersen graphs are cubic graphs, Vizing's theorem implies that their chromatic index is either 3 or 4. The results of this BSc thesis represent an important part of the proof, that the famous Petersen graph is the only generalized Petersen graph, which is not 3-edge colorable. In other words, the Petersen graph GP(5,2) is the only generalized Petersen graph, whose chromatic index equals 4.

• Keywords: mathematics, matematika