Lastnosti grafov Hanojskega stolpa

Zmazek, Eva

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Title:	Lastnosti grafov Hanojskega stolpa
Authors:	ID Zmazek, Eva (Author) ID Klavžar, Sandi (Mentor) More about this mentor...
Files:	MAG_Zmazek_Eva_2019.pdf (554,90 KB) MD5: F41C06D194475BCD4D08ED876C4B1D47 PID: 20.500.12556/dkum/cc42130b-e693-415f-962c-388008f83675
Language:	Slovenian
Work type:	Master's thesis/paper
Typology:	2.09 - Master's Thesis
Organization:	FNM - Faculty of Natural Sciences and Mathematics
Abstract:	Hanojski grafi $H_p^n$ , $n \geq 1$ , $p \geq 3$ , so modeli predstavitve problema Hanojskega stolpa z $n$ diski in $p$ nosilci. Njihova rekurzivna konstrukcija vodi do izpeljave nekaterih lastnosti. Kromatično število $\chi(H_p^n)$ Hanojskega grafa $H_p^n$ je na primer enako številu nosilcev $p$ prirejenega problema Hanojskega stolpa, kromatični indeks $\chi'(H_p^n)$ tega Hanojskega grafa pa je enak njegovi maksimalni stopnji vozlišč $\Delta(H_p^n)$ . Vsi Hanojski grafi so Hamiltonovi, $(p-1)$ -povezani, nekateri med njimi so tudi ravninski. \end{sloppypar} \begin{sloppypar} Barvanje povezav $c: E(G) \to [k]$ je mavrica, če za poljubni različni povezavi $e,f \in E(G)$ velja $c(e) \not= c(f)$. Anti-Ramseyevo število na paru grafov $G$ in $H$ je najmanjše tako število $n$, za katerega pri vsakem barvanju $c$ povezav grafa $G$ z natanko $n$ barvami, obstaja $H$-podgraf grafa $G$, za katerega je zožitev $c\|H$ mavrica. V magistrski nalogi si ogledamo anti-Ramseyeva števila $\ar(H_p^n,H_q^m)$, $p,q \geq 3$, $n,m \geq 1$, na paru Hanojskih grafov, kjer je $m=n=1$ in $q=3$, in na paru Hanojskih grafov, kjer je $p=q$. Za anti-Ramseyevo število $\ar(H_p^n,H_3^1)$, $p \geq 3$, $n \geq 1$, izpeljemo rekurzivno zvezo. Pokažemo tudi, da je anti-Ramseyevo število $\ar(H_4^2,H_3^2)$ omejeno navzdol s $30$ ter navzgor s $34$.
Keywords:	Hanojski graf, Hanojski stolp, anti-Ramseyevo število, mavrica
Place of publishing:	Maribor
Publisher:	[E. Zmazek]
Year of publishing:	2019
PID:	20.500.12556/DKUM-74115
UDC:	519.17(043.2)
COBISS.SI-ID:	24867592
NUK URN:	URN:SI:UM:DK:WB8OWLPT
Publication date in DKUM:	05.11.2019
Views:	1048
Downloads:	121
Metadata:
Categories:	FNM
:	ZMAZEK, Eva, 2019, Lastnosti grafov Hanojskega stolpa [online]. Master’s thesis. Maribor : E. Zmazek. [Accessed 16 March 2025]. Retrieved from: https://dk.um.si/IzpisGradiva.php?lang=eng&id=74115 Copy citation

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Licences

License:	CC BY-NC-ND 4.0, Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International

Link:	http://creativecommons.org/licenses/by-nc-nd/4.0/
Description:	The most restrictive Creative Commons license. This only allows people to download and share the work for no commercial gain and for no other purposes.
Licensing start date:	07.08.2019

Secondary language

Language:	English
Title:	Properties of the graphs of the Tower of Hanoi
Abstract:	For integers $n \geq 1$ and $p \geq 3$ we define Hanoi graph $H_p^n$ as a graph model of Tower of Hanoi with $n$ discs and $p$ pegs. Because of their recursive construction, there are some nice properties of Hanoi graphs. For example, the chromatic number $\chi(H_p^n)$ of Hanoi graph $H_p^n$ is equal to the number of pegs $p$ and the chromatic index $\chi'(H_p^n)$ of the same Hanoi graph is equal to its maximum degree of a vertex, $\Delta(H_p^n)$ . Each Hanoi graph $H_p^n$ is Hamiltonian and $(p-1)$ -connected, and some of them are also planar. Edge coloring $c: E(G) \to [k]$ of graph $G$ is a rainbow if all of its edges are colored with different colors. Anti-Ramsey number for a pair of graphs $G$ and $H$ is the lowest number $n$ such that for every edge coloring $c$ of graph $G$ with exactly $n$ colors there exists such $H$ -subgraph of graph $G$ that the coloring $c$ on it is a rainbow. In the thesis, we present the exact value of anti-Ramsey numbers $\ar(H_p^n,H_q^m)$ , $p,q \geq 3$ , $n,m \geq 1$ , for pairs of Hanoi graph where $n=m=1$ , $q=3$ and also for pairs of Hanoi graphs where $p=q$ . For anti-Ramsey number $\ar(H_p^n,H_3^1)$ , $p \geq 3$ , $n \geq 1$ we give the recursive formula. We also show that the exact value of the anti-Ramsey number $\ar(H_4^2,H_3^2)$ is bounded with $30$ and $34$ .
Keywords:	Hanoi graph, Tower of Hanoi, anti-Ramsey number, rainbow

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