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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
n
p
,
n
≥
1
,
p
≥
3
, so modeli predstavitve problema Hanojskega stolpa z
n
diski in
p
nosilci. Njihova rekurzivna konstrukcija vodi do izpeljave nekaterih lastnosti. Kromatično število
χ
(
H
n
p
)
Hanojskega grafa
H
n
p
je na primer enako številu nosilcev
p
prirejenega problema Hanojskega stolpa, kromatični indeks
χ
′
(
H
n
p
)
tega Hanojskega grafa pa je enak njegovi maksimalni stopnji vozlišč
Δ
(
H
n
p
)
. 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
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Vancouver
:
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
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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
≥
1
and
p
≥
3
we define Hanoi graph
H
n
p
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
χ
(
H
n
p
)
of Hanoi graph
H
n
p
is equal to the number of pegs
p
and the chromatic index
χ
′
(
H
n
p
)
of the same Hanoi graph is equal to its maximum degree of a vertex,
Δ
(
H
n
p
)
. Each Hanoi graph
H
n
p
is Hamiltonian and
(
p
−
1
)
-connected, and some of them are also planar. Edge coloring
c
:
E
(
G
)
→
[
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
n
p
,
H
m
q
)
,
p
,
q
≥
3
,
n
,
m
≥
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
n
p
,
H
1
3
)
,
p
≥
3
,
n
≥
1
we give the recursive formula. We also show that the exact value of the anti-Ramsey number
\ar
(
H
2
4
,
H
2
3
)
is bounded with
30
and
34
.
Keywords:
Hanoi graph
,
Tower of Hanoi
,
anti-Ramsey number
,
rainbow
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