On the Planarity of Hanoi Graphs
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A 2-Parametric Generalization of Sierpinski Gasket Graphs
A 2-parametric generalization of Sierpi´nski gasket graphs Marko Jakovac Faculty of Natural Sciences and Mathematics University of Maribor Koroˇska cesta 160, 2000 Maribor, Slovenia [email protected] Abstract Graphs S[n; k] are introduced as the graphs obtained from the Sierpi´nskigraphs S(n; k) by contracting edges that lie in no complete subgraph Kk. The family S[n; k] is a generalization of a previously studied class of Sierpi´nskigasket graphs Sn. Several properties of graphs S[n; k] are studied in particular, hamiltonicity and chromatic number. Key words: Sierpi´nskigraphs; Sierpi´nskigasket graphs; Hamiltonicity; Chromatic number AMS subject classification (2000): 05C15, 05C45 1 Introduction Sierpi´nski-like graphs appear in many different areas of graph theory, topol- ogy, probability ([8, 11]), psychology ([16]), etc. The special case S(n; 3) turns out to be only a step away of the famous Sierpi´nskigasket graphs Sn|the graphs obtained after a finite number of iterations that in the limit give the Sierpi´nskigasket, see [10]. This connection was introduced by Grundy, Scorer and Smith in [23] and later observed in [24]. Graphs S(n; 3) are also important for the Tower of Hanoi game since they are isomorphic to Hanoi graphs with n discs and 3 pegs. Metric properties, planarity, vertex and edge coloring were studied by now, see, for instance [1, 5, 6, 7, 14, 22]. Furthermore, in [12] it is proved that graphs S(n; 3) are uniquely 3-edge-colorable and have unique Hamiltonian cycles. Graphs S(n; 3) can be generalized to Sierpi´nski graphs S(n; k), k ≥ 3, which are also called Klavˇzar-Milutinovi´cgraphs and denoted KMnk 1 [17]. -
On the Planarity and Hamiltonicity of Hanoi Graphs
ON THE PLANARITY AND HAMILTONICITY OF HANOI GRAPHS Katherine Rock Under the direction of Dr. John Caughman with second reader Dr. Derek Garton MTH 501 Research Project Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Science at Portland State University 2016 Contents 1 Overview 3 1.1 The Tower of Hanoi Puzzles . 3 2 Preliminaries 4 2.1 Graphs . 4 2.2 Hamiltonicity of Graphs . 5 2.3 Planarity of Graphs . 6 2.4 The Hanoi Graphs . 7 2.5 Basic Properties of Hanoi Graphs . 9 3 Hamiltonicity of Hanoi Graphs 10 4 Planarity of Hanoi Graphs 13 2 3 1 Overview The Tower of Hanoi puzzle was created by French number theorist Edouard´ Lucas in 1883, and consists of eight wooden discs of varying size, and three vertical pegs on a wooden base. The legend of the origin of the puzzle may have been created by Lucas, or the legend may have been his inspiration for creating the game. As the legend goes, in the Kashi Vishvanath Temple in the Indian city of Varanasi, beneath a dome that marks the center of the world, there rests a brass plate with three diamond needles. At the beginning of the world, 64 gold discs were placed on one of the needles, the largest resting on the brass plate and the others placed in order of decreasing size from bottom to top. Brahmin priests have been moving the discs day and night since the beginning of time. As the discs are fragile, the priests may not place any larger disc atop any smaller disc. -
Hints and Solutions to Exercises
Appendix A Hints and Solutions to Exercises Chapter 0 since the Fundamental theorem of recur- n 2 sion is of outstanding importance even to Exercise 0.1 The ansatz leads to ξ + n 1 n 2 = define such elementary things like addition ξ + ξ , such that ξ ξ 1, which is solved + = + of the natural numbers, we will state and by τ 1 √5 2 0 and σ 1 √5 2 ∶= ( + )~ > ∶= ( − )~ = prove it here for the connoisseur (and all 1 τ 0. − ~ < n n those who want to become one). Its first ap- Let Fn aτ bσ with 0 F0 a b = + = = + plication will be the definition of factorials and 1 F1 aτ bσ a b √5 2. = = + = ( − ) ~ in the next exercise. Hence a 1 √5 b and consequently τ n =σn ~ = − Fundamental Theorem of Recursion. Let Fn − . (This formula is named for N0 M = √5 M be a set, η0 M, and ϕ M × . Then n n ∈ ∈ N0 J. P. M. Binet.) Similarly, if Ln aτ bσ there exists exactly one mapping η M = + ∈ with 2 L0 a b and 1 L1 1 a which fulfills the recurrence = = + n n= = + ( − b √5 2, we get Ln τ σ . ) ~ = + 0 N0 For n N, we have η 0 η , k η k 1 ϕ k, η k . ∈ ( ) = ∀ ∈ ∶ ( + ) = ( ( )) N0 n 1 n 1 σ n Proof. Uniqueness. Let η, µ M both Fn 1 τ + σ + τ σ τ ∈ + − . fulfill the recurrence. Then η 0 η0 µ 0 n − n σ n Fn = τ σ = ( ) = = ( ) 1 τ and if η k µ k for k N0, then η k 1 − − ( ) = ( ) ∈ ( + ) = 2 ϕ k, η k ϕ k, µ k µ k 1 . -
Jennifer Beineke & Jason Rosenhouse
The Mathematics of Various Entertaining Subjects THE MATHEMATICS OF VARIOUS ENTERTAINING SUBJECTS Volume 2 RESEARCH IN GAMES, GRAPHS, COUNTING, AND COMPLEXITY EDITED BY Jennifer Beineke & Jason Rosenhouse WITH A FOREWORD BY RON GRAHAM National Museum of Mathematics, New York • Princeton University Press, Princeton and Oxford Copyright c 2017 by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 6 Oxford Street, Woodstock, Oxfordshire OX20 1TR press.princeton.edu In association with the National Museum of Mathematics, 11 East 26th Street, New York, New York 10010 Jacket art: Top row (left to right) Fig. 1: Courtesy of Eric Demaine and William S. Moses. Fig. 2: Courtesy of Aviv Adler, Erik Demaine, Adam Hesterberg, Quanquan Liu, and Mikhail Rudoy. Fig. 3: Courtesy of Peter Winkler. Middle row (left to right) Fig. 1: Courtesy of Erik D. Demaine, Martin L. Demaine, Adam Hesterberg, Quanquan Liu, Ron Taylor, and Ryuhei Uehara. Fig. 2: Courtesy of Robert Bosch, Robert Fathauer, and Henry Segerman. Fig. 3: Courtesy of Jason Rosenhouse. Bottom row (left to right) Fig. 1: Courtesy of Noam Elkies. Fig. 2: Courtesy of Richard K. Guy. Fig. 3: Courtesy of Jill Bigley Dunham and Gwyn Whieldon. Excerpt from “Macavity: The Myster Cat” from Old Possum’s Book of Cats by T. S. Eliot. Copyright 1939 by T. S. Eliot. Copyright c Renewed 1967 by Esme Valerie Eliot. Reprinted by permission of Houghton Mifflin Harcourt Publishing Company and Faber & Faber Ltd. All rights reserved. All Rights Reserved Library of Congress Cataloging-in-Publication Data Names: Beineke, Jennifer Elaine, 1969– editor. -
A Survey and Classification of Sierpinski-Type Graphs
A survey and classification of Sierpi´nski-type graphs Andreas M. Hinz a;b Sandi Klavˇzar c;d;b Sara Sabrina Zemljiˇc e;b 2016{02{19 a Mathematical Institute, LMU Munich, Germany [email protected] b Institute of Mathematics, Physics and Mechanics, Ljubljana, Slovenia c Faculty of Mathematics and Physics, University of Ljubljana, Slovenia [email protected] d Faculty of Natural Sciences and Mathematics, University of Maribor, Slovenia e Science Institute, University of Iceland, Reykjavik, Iceland [email protected] Abstract The purpose of this survey is to bring some order into the growing litera- ture on a type of graphs which emerged in the past couple of decades under a wealth of names and in various disguises in different fields of mathematics and its applications. The central role is played by Sierpi´nskigraphs, but we will also shed some light on variants of these graphs and in particular propose their classification. Concentrating on Sierpi´nskigraphs proper we present results on their metric aspects, domination-type invariants with an emphasis on perfect codes, different colorings, and embeddings into other graphs. Keywords: Sierpi´nskitriangle; Sierpi´nskigraphs; Hanoi graphs; graph distance; domination in graphs; graph colorings AMS Subj. Class. (2010): 05-02, 05C12, 05C15, 05C69, 05C75, 05C85 1 Contents 0 Introduction and classification of Sierpi´nski-type graphs 2 0.1 Hanoi and Sierpi´nskigraphs . .2 0.2 Sierpi´nski-type graphs . .8 0.2.1 Sierpi´nskitriangle graphs . 11 0.2.2 Sierpi´nski-like graphs . 15 1 Basic properties of Sierpi´nskigraphs 18 2 Metric properties of Sierpi´nskigraphs 23 2.1 Distance between vertices .