# Analyzing the Smarts Pyramid Puzzle

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Analyzing the Smarts Pyramid Puzzle Spring 2016 Analyzing the Smarts Pyramid Puzzle 1/ 22

2 Outline 1 Defining the problem and its solutions 2 Introduce and prove Burnside s Lemma 3 Proving the results Analyzing the Smarts Pyramid Puzzle 2/ 22

3 The Smarts Pyramid 20 balls 10 pairs 5 colors No two adjacent balls share the same color Analyzing the Smarts Pyramid Puzzle 3/ 22

4 The Smarts Pyramid Definition: A graph G = (V, E) is an ordered pair where V and E are the set of vertices and edges of G, respectively. Defintion: A graph G = (V, E ) is a subgraph of G = (V, E) if V V and E E. Notation: Vertex v = (a, b, c, d) Analyzing the Smarts Pyramid Puzzle 4/ 22

5 Solutions: Matchings Definition: A matching M of a graph G is a subgraph of G such that the edges are pairwise disjoint from each other. Definition: A graph with an edge between every pair of two vertices is a complete graph. Definition: A complete graph with n vertices is a K n graph. Analyzing the Smarts Pyramid Puzzle 5/ 22

6 Matchings:T i -matchings Definition: A tetrahedron is a K 4 subgraph of G. Definition: A matching M of graph G contains a tetrahedron if M contains two edges whose corresponding vertices in G are adjacent and form a K 4 subgraph of G. Definition: A T i -matching is a matching M where i N is the number of tetrahedra contained in M. Analyzing the Smarts Pyramid Puzzle 6/ 22

7 Solutions: Colorings Definition: A k-coloring of the graph P is an assignment of colors represented by integers such that no pair of adjacent vertices share a common color. Analyzing the Smarts Pyramid Puzzle 7/ 22

8 Equivalencies and Symmetries Given the coordinates (a, b, c, d), 24 permutations can be applied to it. The 24 permutations consist of: Single Reflections: (a, b, c, d) (b, a, c, d). Double Reflections: (a, b, c, d) (b, a, d, c). Rotations: (a, b, c, d) (a, d, b, c). Four-Cycles: (a, b, c, d) (b, c, d, a). Identity: (a, b, c, d) (a, b, c, d) Permutation g on element x is denoted as g x. Analyzing the Smarts Pyramid Puzzle 8/ 22

9 Equivalencies and Symmetries Definition: Two colorings, C and C are equivalent if and only if there exists a permutation g such that for every vertex v P, C(g v) = C (v). Two matchings M = (V, E) and M = (V, E ) are equivalent if and only if there exists a permutation g such that for all vertices v in V, g v V. Definition: A coloring C is fixed by g if and only if for every vertex v of P, C(g v) = C(v). Analyzing the Smarts Pyramid Puzzle 9/ 22

10 Burnside s Lemma Given a Group G that acts on a set S: Definition: An orbit of x under Group G is denoted by orb G (x) = {g x g G} Definition: A stabilizer of x under Group G is denoted by stab G (x) = {g G g x = x} Definition: The number of fixed points of g is denoted by fix(g). fix(g) = {x S g x = x} The Orbit-Stabilizer Theorem: Given finite group G which acts on a finite set S, for each x S, stab G (x) orb G (x) = G. Analyzing the Smarts Pyramid Puzzle 10/ 22

11 Burnside s Lemma Burnside s Lemma: Given a finite group G that acts on a finite set S. G X X The number of orbits in set S is equal to the sum over all g G of the cardinality of the subsets of S where each subset is fixed by some g. number of orbits = 1 G g G fix(g) Analyzing the Smarts Pyramid Puzzle 11/ 22

12 Burnside s Lemma: Proof Step 1. x S stab G (x) = g G fix(g) Consider a matrix A with rows indexed by x S and with columns indexed by g G. A will be defined such that A xg { 1, g x = x 0, otherwise Given x S, the sum across a row is stab G (x). And, given a g G, the sum down a column is fix(g). x S stab G (x) = x S (sum of all 1 s in row x) = the total number of 1 s in the matrix. g G fix(g) = g G (sum of all 1 s in column g) = the total number of 1 s in the matrix. Thus, x S stab G (x) = g G fix(g), as wanted. Analyzing the Smarts Pyramid Puzzle 12/ 22

13 Burnside s Lemma: Proof Step 2. Orbit Stabilizer Theorem The Orbit-Stabilizer Theorem can be rewritten from stab G (x) orb G (x) = G. stab G (x) G to = 1 orb G (x) Analyzing the Smarts Pyramid Puzzle 13/ 22

14 Burnside s Lemma: Proof Step 3. Putting it together x S stab G (x) = g G fix(g) from Step 1. stab G (x) = fix(g) g G 1 G x S x S x S stab G (x) = 1 fix(g) G g G 1 G stab G (x) = 1 fix(g) G g G Use the Orbit Stabilizer Theorem to get: x S 1 orb G (x) = 1 fix(g) G g G Analyzing the Smarts Pyramid Puzzle 14/ 22

15 Burnside s Lemma: Proof x S 1 orb G (x) = 1 fix(g) G g G Focusing on the left side of the equation: Rewrite sum as the double sum orbits x orbit. orbits x orbit 1 orb G (x) = 1 fix(g) G g G Inner Sum: Given the orbit {x 1, x 2,..., x t }, each x {x 1, x 2,..., x t } has an orbit, orb G (x), of size t. Thus, we get t i=1 1 t = 1. orbits (1) = 1 fix(g) G g G Analyzing the Smarts Pyramid Puzzle 15/ 22

16 Burnside s Lemma: Proof Thus, number of orbits = 1 fix(g) G g G Analyzing the Smarts Pyramid Puzzle 16/ 22

17 Results: T 5 -matchings Three ways to construct a matching for a tetrahedron. With 5 tetrahedra: 3 5 = distinct T 5 -matchings up to equivalence Analyzing the Smarts Pyramid Puzzle 17/ 22

18 T 5 -matchings: Burnside s Lemma Table: Burnside s Lemma: T 5 -matchings Action Number Matchings Fixed Total Single Reflection Double Reflection Rotation Cycle Identity Burnside s Lemma: number of orbits = 1 G fix(g) g G number of orbits = 1 24 ( ) number of orbits = = 12 Analyzing the Smarts Pyramid Puzzle 18/ 22

19 Results: Colorings ( 20 4 ) ( 16 ) ( 4 12 ) ( 4 8 ) ( 4 4 ) 4 = 305, 540, 235, 000 ways to assign colors to vertices 3778 colorings 183 distinct colorings up to equivalence Analyzing the Smarts Pyramid Puzzle 19/ 22

20 Colorings: Burnside s Lemma Burnside s Lemma: Table: Burnside s Lemma: Colorings orb S4 (x) stab S4 (x) Occurrences fix(g) omitted number of orbits = 1 G fix(g) g G number of orbits = 1 24 ( ) number of orbits = = 183 Analyzing the Smarts Pyramid Puzzle 20/ 22

21 Results The Un-Named Theorem: Each coloring admits exactly 6 T 5 -matchings. Each of the 183 colorings have 1 to 6 non-isomorphic T 5 -matchings. There are between 183 and 1098 unique solutions to the Smarts Pyramid using T 5 -matchings. Analyzing the Smarts Pyramid Puzzle 21/ 22

22 References Bondy, J.A., and U. S. R. Murty. Graduate Texts in Mathematics: Graph Theory. N.p.: Springer, Print Klingsberg, Paul. Group Actions and Burnside s Lemma. Saint Joseph s University, n.d. Web. 3 Nov < pklingsb/burnside> Multiplication Table of S4. S4 Tables. Bogazici University, n.d. Web. 17 Oct < Roberts, Fred S., and Barry Tesman. Applied Combinatorics. CRC Press, Print Saracino, Dan. Abstract Algebra: A First Course. Reading, MA: Addison-Wesley, Print. Analyzing the Smarts Pyramid Puzzle 22/ 22

23 Equivalencies and Symmetries Definition: A symmetry group of degree n, denoted by S n, is the group of all permutations g on a finite set of n elements. Analyzing the Smarts Pyramid Puzzle 23/ 22

24 Theorem 1 Theorem: There is only one coloring, up to equivalence, that is symmetric over all permuations. Definition: Two colors c 1 and c 2 are interchangeable if all vertices colored c 1 can be taken to vertices colored c 2 by a graph automorphism, i.e., by recoloring c 1 to c 2 and c 2 to c 1, we get an equivalent coloring. Assume P is in a T 5 -matching. Without loss of generality, the center tetrahedron is colored 0, 1, 2, and 3. 0, 1, 2, and 3 are interchangeable. Analyzing the Smarts Pyramid Puzzle 24/ 22

25 Theorem 1 c3 c 0 = 0 c 1 = 1 c 2 = 2 c 3 = 3 c0 c2 c1 Analyzing the Smarts Pyramid Puzzle 25/ 22

26 Theorem 1 4 is not interchangeable with any other color, thus any symmetry takes 4 to itself. The four corners of the pyramid are colored 4. c3 v3 c2 v1 v4 c1 v2 Analyzing the Smarts Pyramid Puzzle 26/ 22

27 Theorem 1 Vertices v 1, v 2 and v 3 each have one option for a valid coloring. c 2 and v 4 are adjacent to v 1, v 2, and v 3, thus v 1, v 2, and v 3 may not be colored 2 or 4. They may also not share a color. v 1 is adjacent to c 1, c 2, and c 3, thus the only valid color for v 1 is 0. v 3 is adjacent to c 3, which is colored 3. So, v 3 must be colored 1. v 2 is colored 3 Analyzing the Smarts Pyramid Puzzle 27/ 22

28 Theorem 1 Analyzing the Smarts Pyramid Puzzle 28/ 22

29 Theorem 2 Theorem: Any coloring of the Smarts Pyramid admits exactly six T 5 -matchings. 1. Any coloring of the pyramid has at most six T 5 -matchings. 2. Any coloring of the pyramid has at least six T 5 -matchings. Analyzing the Smarts Pyramid Puzzle 29/ 22

30 Theorem 2: Part 1 Step 1. i j, the colorings of T i and T j do not contain exactly the same colors. Given four of each color (1, 2, 3, 4, and 5), we have q = [1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5] where q is a list representing available color assignments. Assume towards contradiction that there exists T 1 and T 2 such that their colorings both contain the colors 1, 2, 3, and 4. The remaining available color assignments are then q = [1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 5, 5]. By the pigeonhole principle, at least one tetrahedron will have two 5 s in its coloring. Thus, i j, T i and T j have different colorings. Analyzing the Smarts Pyramid Puzzle 30/ 22

31 Theorem 2: Part 1 Step 2. There is only one way to color five tetrahedra such that no two tetrahedra have the same coloring. A tetrahedron is made up of four different colors. With 5 colors to choose from, the number of ways to color a tetrahedron is ( 5 4) = 5. Because of we must color 5 tetrahedra, there is only one way to uniquely color 5 tetrahedra. Analyzing the Smarts Pyramid Puzzle 31/ 22

32 Theorem 2: Part 1 Step 3. Given the colorings in Step 2, there are six T 5 -matchings compatible with it. Notation: An edge whose vertices are colored will be denoted by [a, b], where a and b represents the vertices colors. The union of two edges is a matching of a tetrahedron. Any given edge [a, b] can be paired with three other edges to make a matching of a colored tetrahedron. Without loss of generality, if [1,2] is paired with the edge [3,4], then the edges that contain a vertex colored 5, or [*,5], are [1,5], [2,5], [3,5], and [4,5]. The other remaining edges are: [1,3], [1,4], [2,3], and [2,4]. No edge [*, 5] can be paired with another [*,5]. So, each [*,5] is paired with an edge that does not contain a vertex which is colored 5, otherwise, by the pigeonhole principle, a [*,5] will be paired with another [*,5]. Each edge without a five can be paired with exactly two [*,5]. Analyzing the Smarts Pyramid Puzzle 32/ 22

33 Theorem 2: Part 1 Each edge without a five can be paired with exactly two [*,5]. Edge [1,3] can be paired with either [2,5] or [4,5]. If [1,3] is paired with [2,5], then [1,4] must be paired with [3,5] Then [2,4] must be paired with [1,5], and [2,3] with [4,5]. Likewise, if [1,3] is paired with [4,5] instead of [2,5], then the remaining matchings must be: [2, 3] to [1, 5] [2, 4] to [3, 5] [1, 4] to [2, 5] The first tetrahedron matched has three matchings to choose from. The remaining eight edges can be paired in two different ways. So, there are at most six matchings for five tetrahedra that each have a different coloring. Analyzing the Smarts Pyramid Puzzle 33/ 22

34 Theorem 2: Part 2 From the set of five colored tetrahedra of P, choose one tetrahedron, let s say T 1 that has no matching assigned. This tetrahedron s coloring, without loss of generality, includes the colors 1, 2, 3, and 4. This means that the coloring of the other four tetrahedra include the colors as follows: The coloring of T 2 includes the colors 1, 2, 3, and 5 The coloring of T 3 includes the colors 1, 2, 4, and 5 The coloring of T 4 includes the colors 1, 3, 4, and 5 The coloring of T 5 includes the colors 2, 3, 4, and 5 Analyzing the Smarts Pyramid Puzzle 34/ 22

35 Theorem 2: Part 2 In the matching of T 1, the vertex colored 1 can be in an edge with the vertex colored 2, 3, or 4. Assume that the matching of T 1 is made of of edges [1, 2] and [3, 4]. T 2 and T 3 also contain the colors 1 and 2. T 2 also contains 3 and 5. T 3 also contains 4 and 5. T 1, T 2, and T 3 are equivalent up to recoloring. T 1 contains [3,4], so T 4 must have [3,5] or [4,5] If T 4 has [3,5], then T 5 has [4,5] and if T 4 has [4,5], then T 5 has [3,5]. From the three choices for matching T 1 and the two choices for matching T 4 and T 5, we conclude that there are at least six ways to match the five tetrahedra. Analyzing the Smarts Pyramid Puzzle 35/ 22

### ENUMERATION BY ALGEBRAIC COMBINATORICS. 1. Introduction

ENUMERATION BY ALGEBRAIC COMBINATORICS CAROLYN ATWOOD Abstract. Pólya s theorem can be used to enumerate objects under permutation groups. Using group theory, combinatorics, and many examples, Burnside

### COMBINATORIAL PROPERTIES OF THE HIGMAN-SIMS GRAPH. 1. Introduction

COMBINATORIAL PROPERTIES OF THE HIGMAN-SIMS GRAPH ZACHARY ABEL 1. Introduction In this survey we discuss properties of the Higman-Sims graph, which has 100 vertices, 1100 edges, and is 22 regular. In fact

### GROUPS ACTING ON A SET

GROUPS ACTING ON A SET MATH 435 SPRING 2012 NOTES FROM FEBRUARY 27TH, 2012 1. Left group actions Definition 1.1. Suppose that G is a group and S is a set. A left (group) action of G on S is a rule for

### Zachary Monaco Georgia College Olympic Coloring: Go For The Gold

Zachary Monaco Georgia College Olympic Coloring: Go For The Gold Coloring the vertices or edges of a graph leads to a variety of interesting applications in graph theory These applications include various

### Notes on Matrix Multiplication and the Transitive Closure

ICS 6D Due: Wednesday, February 25, 2015 Instructor: Sandy Irani Notes on Matrix Multiplication and the Transitive Closure An n m matrix over a set S is an array of elements from S with n rows and m columns.

### On Integer Additive Set-Indexers of Graphs

On Integer Additive Set-Indexers of Graphs arxiv:1312.7672v4 [math.co] 2 Mar 2014 N K Sudev and K A Germina Abstract A set-indexer of a graph G is an injective set-valued function f : V (G) 2 X such that

### Complete graphs whose topological symmetry groups are polyhedral

Algebraic & Geometric Topology 11 (2011) 1405 1433 1405 Complete graphs whose topological symmetry groups are polyhedral ERICA FLAPAN BLAKE MELLOR RAMIN NAIMI We determine for which m the complete graph

### CLASSIFYING FINITE SUBGROUPS OF SO 3

CLASSIFYING FINITE SUBGROUPS OF SO 3 HANNAH MARK Abstract. The goal of this paper is to prove that all finite subgroups of SO 3 are isomorphic to either a cyclic group, a dihedral group, or the rotational

### Geometric Group Theory Homework 1 Solutions. e a b e e a b a a b e b b e a

Geometric Group Theory Homework 1 Solutions Find all groups G with G = 3 using multiplications tables and the group axioms. A multiplication table for a group must be a Latin square: each element appears

### arxiv: v1 [math.gr] 18 Apr 2016

Unfoldings of the Cube Richard Goldstone Manhattan College Riverdale, NY 1071 richard.goldstone@manhattan.edu arxiv:160.0500v1 [math.gr] 18 Apr 2016 Robert Suzzi Valli Manhattan College Riverdale, NY 1071

### A CHARACTERIZATION OF TREE TYPE

A CHARACTERIZATION OF TREE TYPE LON H MITCHELL Abstract Let L(G) be the Laplacian matrix of a simple graph G The characteristic valuation associated with the algebraic connectivity a(g) is used in classifying

### Solutions to Exercises 8

Discrete Mathematics Lent 2009 MA210 Solutions to Exercises 8 (1) Suppose that G is a graph in which every vertex has degree at least k, where k 1, and in which every cycle contains at least 4 vertices.

### International Electronic Journal of Pure and Applied Mathematics IEJPAM, Volume 9, No. 1 (2015)

International Electronic Journal of Pure and Applied Mathematics Volume 9 No. 1 2015, 21-27 ISSN: 1314-0744 url: http://www.e.ijpam.eu doi: http://dx.doi.org/10.12732/iejpam.v9i1.3 THE ISOTROPY GRAPH OF

### Math 4707: Introduction to Combinatorics and Graph Theory

Math 4707: Introduction to Combinatorics and Graph Theory Lecture Addendum, November 3rd and 8th, 200 Counting Closed Walks and Spanning Trees in Graphs via Linear Algebra and Matrices Adjacency Matrices

### Clique coloring B 1 -EPG graphs

Clique coloring B 1 -EPG graphs Flavia Bonomo a,c, María Pía Mazzoleni b,c, and Maya Stein d a Departamento de Computación, FCEN-UBA, Buenos Aires, Argentina. b Departamento de Matemática, FCE-UNLP, La

### Magic Hexagon Puzzles

Magic Hexagon Puzzles Arthur Holshouser 3600 Bullard St Charlotte NC USA Harold Reiter Department of Mathematics University of North Carolina Charlotte Charlotte NC 28223 USA hbreiter@unccedu 1 Abstract

### The chromatic spectrum of mixed hypergraphs

The chromatic spectrum of mixed hypergraphs Tao Jiang, Dhruv Mubayi, Zsolt Tuza, Vitaly Voloshin, Douglas B. West March 30, 2003 Abstract A mixed hypergraph is a triple H = (X, C, D), where X is the vertex

### 13 Solutions for Section 6

13 Solutions for Section 6 Exercise 6.2 Draw up the group table for S 3. List, giving each as a product of disjoint cycles, all the permutations in S 4. Determine the order of each element of S 4. Solution

### S on n elements. A good way to think about permutations is the following. Consider the A = 1,2,3, 4 whose elements we permute with the P =

Section 6. 1 Section 6. Groups of Permutations: : The Symmetric Group Purpose of Section: To introduce the idea of a permutation and show how the set of all permutations of a set of n elements, equipped

### THE 0/1-BORSUK CONJECTURE IS GENERICALLY TRUE FOR EACH FIXED DIAMETER

THE 0/1-BORSUK CONJECTURE IS GENERICALLY TRUE FOR EACH FIXED DIAMETER JONATHAN P. MCCAMMOND AND GÜNTER ZIEGLER Abstract. In 1933 Karol Borsuk asked whether every compact subset of R d can be decomposed

### SHORT CYCLE COVERS OF GRAPHS WITH MINIMUM DEGREE THREE

SHOT YLE OVES OF PHS WITH MINIMUM DEEE THEE TOMÁŠ KISE, DNIEL KÁL, END LIDIKÝ, PVEL NEJEDLÝ OET ŠÁML, ND bstract. The Shortest ycle over onjecture of lon and Tarsi asserts that the edges of every bridgeless

### Odd induced subgraphs in graphs of maximum degree three

Odd induced subgraphs in graphs of maximum degree three David M. Berman, Hong Wang, and Larry Wargo Department of Mathematics University of New Orleans New Orleans, Louisiana, USA 70148 Abstract A long-standing

### Braids and Juggling Patterns. Matthew Macauley Michael Orrison, Advisor

Braids and Juggling Patterns by Matthew Macauley Michael Orrison, Advisor Advisor: Second Reader: (Jim Hoste) May 2003 Department of Mathematics Abstract Braids and Juggling Patterns by Matthew Macauley

### 10. Graph Matrices Incidence Matrix

10 Graph Matrices Since a graph is completely determined by specifying either its adjacency structure or its incidence structure, these specifications provide far more efficient ways of representing a

### GROUP ACTIONS ON SETS WITH APPLICATIONS TO FINITE GROUPS

GROUP ACTIONS ON SETS WITH APPLICATIONS TO FINITE GROUPS NOTES OF LECTURES GIVEN AT THE UNIVERSITY OF MYSORE ON 29 JULY, 01 AUG, 02 AUG, 2012 K. N. RAGHAVAN Abstract. The notion of the action of a group

### Characterizations of Arboricity of Graphs

Characterizations of Arboricity of Graphs Ruth Haas Smith College Northampton, MA USA Abstract The aim of this paper is to give several characterizations for the following two classes of graphs: (i) graphs

### 2.3 Scheduling jobs on identical parallel machines

2.3 Scheduling jobs on identical parallel machines There are jobs to be processed, and there are identical machines (running in parallel) to which each job may be assigned Each job = 1,,, must be processed

### Outline 2.1 Graph Isomorphism 2.2 Automorphisms and Symmetry 2.3 Subgraphs, part 1

GRAPH THEORY LECTURE STRUCTURE AND REPRESENTATION PART A Abstract. Chapter focuses on the question of when two graphs are to be regarded as the same, on symmetries, and on subgraphs.. discusses the concept

### 1. Relevant standard graph theory

Color identical pairs in 4-chromatic graphs Asbjørn Brændeland I argue that, given a 4-chromatic graph G and a pair of vertices {u, v} in G, if the color of u equals the color of v in every 4-coloring

### Relations Graphical View

Relations Slides by Christopher M. Bourke Instructor: Berthe Y. Choueiry Introduction Recall that a relation between elements of two sets is a subset of their Cartesian product (of ordered pairs). A binary

### Edge looseness of plane graphs

Also available at http://amc-journal.eu ISSN 1855-966 (printed edn.), ISSN 1855-974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 9 (015) 89 96 Edge looseness of plane graphs Július Czap Department of

### You ve seen them played in coffee shops, on planes, and

Every Sudoku variation you can think of comes with its own set of interesting open questions There is math to be had here. So get working! Taking Sudoku Seriously Laura Taalman James Madison University

### Discrete Mathematics Problems

Discrete Mathematics Problems William F. Klostermeyer School of Computing University of North Florida Jacksonville, FL 32224 E-mail: wkloster@unf.edu Contents 0 Preface 3 1 Logic 5 1.1 Basics...............................

### Lecture notes from Foundations of Markov chain Monte Carlo methods University of Chicago, Spring 2002 Lecture 1, March 29, 2002

Lecture notes from Foundations of Markov chain Monte Carlo methods University of Chicago, Spring 2002 Lecture 1, March 29, 2002 Eric Vigoda Scribe: Varsha Dani & Tom Hayes 1.1 Introduction The aim of this

### FRACTIONAL COLORINGS AND THE MYCIELSKI GRAPHS

FRACTIONAL COLORINGS AND THE MYCIELSKI GRAPHS By G. Tony Jacobs Under the Direction of Dr. John S. Caughman A math 501 project submitted in partial fulfillment of the requirements for the degree of Master

### Trees and Fundamental Circuits

Trees and Fundamental Circuits Tree A connected graph without any circuits. o must have at least one vertex. o definition implies that it must be a simple graph. o only finite trees are being considered

### DO NOT RE-DISTRIBUTE THIS SOLUTION FILE

Professor Kindred Math 04 Graph Theory Homework 7 Solutions April 3, 03 Introduction to Graph Theory, West Section 5. 0, variation of 5, 39 Section 5. 9 Section 5.3 3, 8, 3 Section 7. Problems you should

### Tenacity and rupture degree of permutation graphs of complete bipartite graphs

Tenacity and rupture degree of permutation graphs of complete bipartite graphs Fengwei Li, Qingfang Ye and Xueliang Li Department of mathematics, Shaoxing University, Shaoxing Zhejiang 312000, P.R. China

Some Definitions about Sets Definition: Two sets are equal if they contain the same elements. I.e., sets A and B are equal if x[x A x B]. Notation: A = B. Recall: Sets are unordered and we do not distinguish

### Partitioning edge-coloured complete graphs into monochromatic cycles and paths

arxiv:1205.5492v1 [math.co] 24 May 2012 Partitioning edge-coloured complete graphs into monochromatic cycles and paths Alexey Pokrovskiy Departement of Mathematics, London School of Economics and Political

### Pigeonhole Principle Solutions

Pigeonhole Principle Solutions 1. Show that if we take n + 1 numbers from the set {1, 2,..., 2n}, then some pair of numbers will have no factors in common. Solution: Note that consecutive numbers (such

### vertex, 369 disjoint pairwise, 395 disjoint sets, 236 disjunction, 33, 36 distributive laws

Index absolute value, 135 141 additive identity, 254 additive inverse, 254 aleph, 466 algebra of sets, 245, 278 antisymmetric relation, 387 arcsine function, 349 arithmetic sequence, 208 arrow diagram,

### Homework MA 725 Spring, 2012 C. Huneke SELECTED ANSWERS

Homework MA 725 Spring, 2012 C. Huneke SELECTED ANSWERS 1.1.25 Prove that the Petersen graph has no cycle of length 7. Solution: There are 10 vertices in the Petersen graph G. Assume there is a cycle C

### Complexity and Completeness of Finding Another Solution and Its Application to Puzzles

yato@is.s.u-tokyo.ac.jp seta@is.s.u-tokyo.ac.jp Π (ASP) Π x s x s ASP Ueda Nagao n n-asp parsimonious ASP ASP NP Complexity and Completeness of Finding Another Solution and Its Application to Puzzles Takayuki

### Notes on finite group theory. Peter J. Cameron

Notes on finite group theory Peter J. Cameron October 2013 2 Preface Group theory is a central part of modern mathematics. Its origins lie in geometry (where groups describe in a very detailed way the

### Midterm Practice Problems

6.042/8.062J Mathematics for Computer Science October 2, 200 Tom Leighton, Marten van Dijk, and Brooke Cowan Midterm Practice Problems Problem. [0 points] In problem set you showed that the nand operator

### HEX. 1. Introduction

HEX SP.268 SPRING 2011 1. Introduction The game of Hex was first invented in 1942 by Piet Hein, a Danish scientist, mathematician, writer, and poet. In 1948, John Nash at Princeton re-discovered the game,

### Lattice Point Geometry: Pick s Theorem and Minkowski s Theorem. Senior Exercise in Mathematics. Jennifer Garbett Kenyon College

Lattice Point Geometry: Pick s Theorem and Minkowski s Theorem Senior Exercise in Mathematics Jennifer Garbett Kenyon College November 18, 010 Contents 1 Introduction 1 Primitive Lattice Triangles 5.1

### TREES IN SET THEORY SPENCER UNGER

TREES IN SET THEORY SPENCER UNGER 1. Introduction Although I was unaware of it while writing the first two talks, my talks in the graduate student seminar have formed a coherent series. This talk can be

### On the Ramsey Number R(4, 6)

On the Ramsey Number R(4, 6) Geoffrey Exoo Department of Mathematics and Computer Science Indiana State University Terre Haute, IN 47809 ge@cs.indstate.edu Submitted: Mar 6, 2012; Accepted: Mar 23, 2012;

### Construction and Properties of the Icosahedron

Course Project (Introduction to Reflection Groups) Construction and Properties of the Icosahedron Shreejit Bandyopadhyay April 19, 2013 Abstract The icosahedron is one of the most important platonic solids

### Diameter of paired domination edge-critical graphs

AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 40 (2008), Pages 279 291 Diameter of paired domination edge-critical graphs M. Edwards R.G. Gibson Department of Mathematics and Statistics University of Victoria

### Finite and Infinite Sets

Chapter 9 Finite and Infinite Sets 9. Finite Sets Preview Activity (Equivalent Sets, Part ). Let A and B be sets and let f be a function from A to B..f W A! B/. Carefully complete each of the following

### Problem Set 1 Solutions Math 109

Problem Set 1 Solutions Math 109 Exercise 1.6 Show that a regular tetrahedron has a total of twenty-four symmetries if reflections and products of reflections are allowed. Identify a symmetry which is

### Planarity Planarity

Planarity 8.1 71 Planarity Up until now, graphs have been completely abstract. In Topological Graph Theory, it matters how the graphs are drawn. Do the edges cross? Are there knots in the graph structure?

### 1 Digraphs. Definition 1

1 Digraphs Definition 1 Adigraphordirected graphgisatriplecomprisedofavertex set V(G), edge set E(G), and a function assigning each edge an ordered pair of vertices (tail, head); these vertices together

### A set is a Many that allows itself to be thought of as a One. (Georg Cantor)

Chapter 4 Set Theory A set is a Many that allows itself to be thought of as a One. (Georg Cantor) In the previous chapters, we have often encountered sets, for example, prime numbers form a set, domains

### Announcements. CompSci 230 Discrete Math for Computer Science. Test 1

CompSci 230 Discrete Math for Computer Science Sep 26, 2013 Announcements Exam 1 is Tuesday, Oct. 1 No class, Oct 3, No recitation Oct 4-7 Prof. Rodger is out Sep 30-Oct 4 There is Recitation: Sept 27-30.

### 2.3. Relations. Arrow diagrams. Venn diagrams and arrows can be used for representing

2.3. RELATIONS 32 2.3. Relations 2.3.1. Relations. Assume that we have a set of men M and a set of women W, some of whom are married. We want to express which men in M are married to which women in W.

### DETERMINANTS IN THE KRONECKER PRODUCT OF MATRICES: THE INCIDENCE MATRIX OF A COMPLETE GRAPH

DETERMINANTS IN THE KRONECKER PRODUCT OF MATRICES: THE INCIDENCE MATRIX OF A COMPLETE GRAPH CHRISTOPHER RH HANUSA AND THOMAS ZASLAVSKY Abstract We investigate the least common multiple of all subdeterminants,

### Degree diameter problem on triangular networks

AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 63(3) (2015), Pages 333 345 Degree diameter problem on triangular networks Přemysl Holub Department of Mathematics University of West Bohemia P.O. Box 314,

### MODULAR ARITHMETIC. a smallest member. It is equivalent to the Principle of Mathematical Induction.

MODULAR ARITHMETIC 1 Working With Integers The usual arithmetic operations of addition, subtraction and multiplication can be performed on integers, and the result is always another integer Division, on

### Notes on polyhedra and 3-dimensional geometry

Notes on polyhedra and 3-dimensional geometry Judith Roitman / Jeremy Martin April 23, 2013 1 Polyhedra Three-dimensional geometry is a very rich field; this is just a little taste of it. Our main protagonist

### 4 Basics of Trees. Petr Hliněný, FI MU Brno 1 FI: MA010: Trees and Forests

4 Basics of Trees Trees, actually acyclic connected simple graphs, are among the simplest graph classes. Despite their simplicity, they still have rich structure and many useful application, such as in

### Discrete Mathematics & Mathematical Reasoning Chapter 6: Counting

Discrete Mathematics & Mathematical Reasoning Chapter 6: Counting Colin Stirling Informatics Slides originally by Kousha Etessami Colin Stirling (Informatics) Discrete Mathematics (Chapter 6) Today 1 /

### Lecture 6: Approximation via LP Rounding

Lecture 6: Approximation via LP Rounding Let G = (V, E) be an (undirected) graph. A subset C V is called a vertex cover for G if for every edge (v i, v j ) E we have v i C or v j C (or both). In other

### Topological Treatment of Platonic, Archimedean, and Related Polyhedra

Forum Geometricorum Volume 15 (015) 43 51. FORUM GEOM ISSN 1534-1178 Topological Treatment of Platonic, Archimedean, and Related Polyhedra Tom M. Apostol and Mamikon A. Mnatsakanian Abstract. Platonic

### A simple proof for the number of tilings of quartered Aztec diamonds

A simple proof for the number of tilings of quartered Aztec diamonds arxiv:1309.6720v1 [math.co] 26 Sep 2013 Tri Lai Department of Mathematics Indiana University Bloomington, IN 47405 tmlai@indiana.edu

### Actually Doing It! 6. Prove that the regular unit cube (say 1cm=unit) of sufficiently high dimension can fit inside it the whole city of New York.

1: 1. Compute a random 4-dimensional polytope P as the convex hull of 10 random points using rand sphere(4,10). Run VISUAL to see a Schlegel diagram. How many 3-dimensional polytopes do you see? How many

### Sudoku Puzzles and Mathematical Expressions

Sudoku Puzzles and Mathematical Expressions Austin Sloane May 16, 2015 1 List of Figures 1 A blank Sudoku grid........................ 3 2 A solved Sudoku puzzle...................... 4 3 A Latin square

### Aztec Diamonds and Baxter Permutations

Aztec Diamonds and Baxter Permutations Hal Canary Mathematics Department University of Wisconsin Madison h3@halcanary.org Submitted: Jul 9, 29; Accepted: Jul 3, 2; Published: Aug 9, 2 Mathematics Subject

### Lecture 4: The Chromatic Number

Introduction to Graph Theory Instructor: Padraic Bartlett Lecture 4: The Chromatic Number Week 1 Mathcamp 2011 In our discussion of bipartite graphs, we mentioned that one way to classify bipartite graphs

### Matrices, transposes, and inverses

Matrices, transposes, and inverses Math 40, Introduction to Linear Algebra Wednesday, February, 202 Matrix-vector multiplication: two views st perspective: A x is linear combination of columns of A 2 4

### COLORED GRAPHS AND THEIR PROPERTIES

COLORED GRAPHS AND THEIR PROPERTIES BEN STEVENS 1. Introduction This paper is concerned with the upper bound on the chromatic number for graphs of maximum vertex degree under three different sets of coloring

### Basic Notions on Graphs. Planar Graphs and Vertex Colourings. Joe Ryan. Presented by

Basic Notions on Graphs Planar Graphs and Vertex Colourings Presented by Joe Ryan School of Electrical Engineering and Computer Science University of Newcastle, Australia Planar graphs Graphs may be drawn

### Chapter 4. Trees. 4.1 Basics

Chapter 4 Trees 4.1 Basics A tree is a connected graph with no cycles. A forest is a collection of trees. A vertex of degree one, particularly in a tree, is called a leaf. Trees arise in a variety of applications.

### AN ANALYSIS OF THE 15-PUZZLE

AN ANALYSIS OF THE 15-PUZZLE ANDREW CHAPPLE, ALFONSO CROEZE, MHEL LAZO, AND HUNTER MERRILL Abstract. The 15-puzzle has been an object of great mathematical interest since its invention in the 1860s. The

### Graph Isomorphism Completeness for Perfect Graphs and Subclasses of Perfect Graphs

Graph Isomorphism Completeness for Perfect Graphs and Subclasses of Perfect Graphs C. Boucher D. Loker May 2006 Abstract A problem is said to be GI-complete if it is provably as hard as graph isomorphism;

### Spring 2007 Math 510 Hints for practice problems

Spring 2007 Math 510 Hints for practice problems Section 1 Imagine a prison consisting of 4 cells arranged like the squares of an -chessboard There are doors between all adjacent cells A prisoner in one

### GROUPS SUBGROUPS. Definition 1: An operation on a set G is a function : G G G.

Definition 1: GROUPS An operation on a set G is a function : G G G. Definition 2: A group is a set G which is equipped with an operation and a special element e G, called the identity, such that (i) the

### Applied Algorithm Design Lecture 5

Applied Algorithm Design Lecture 5 Pietro Michiardi Eurecom Pietro Michiardi (Eurecom) Applied Algorithm Design Lecture 5 1 / 86 Approximation Algorithms Pietro Michiardi (Eurecom) Applied Algorithm Design

### Chapter 3. Cartesian Products and Relations. 3.1 Cartesian Products

Chapter 3 Cartesian Products and Relations The material in this chapter is the first real encounter with abstraction. Relations are very general thing they are a special type of subset. After introducing

### Math 475, Problem Set #13: Answers. A. A baton is divided into five cylindrical bands of equal length, as shown (crudely) below.

Math 475, Problem Set #13: Answers A. A baton is divided into five cylindrical bands of equal length, as shown (crudely) below. () ) ) ) ) ) In how many different ways can the five bands be colored if

### Cycles in a Graph Whose Lengths Differ by One or Two

Cycles in a Graph Whose Lengths Differ by One or Two J. A. Bondy 1 and A. Vince 2 1 LABORATOIRE DE MATHÉMATIQUES DISCRÉTES UNIVERSITÉ CLAUDE-BERNARD LYON 1 69622 VILLEURBANNE, FRANCE 2 DEPARTMENT OF MATHEMATICS

### The Pigeonhole Principle

The Pigeonhole Principle 1 Pigeonhole Principle: Simple form Theorem 1.1. If n + 1 objects are put into n boxes, then at least one box contains two or more objects. Proof. Trivial. Example 1.1. Among 13

### SEQUENCES OF MAXIMAL DEGREE VERTICES IN GRAPHS. Nickolay Khadzhiivanov, Nedyalko Nenov

Serdica Math. J. 30 (2004), 95 102 SEQUENCES OF MAXIMAL DEGREE VERTICES IN GRAPHS Nickolay Khadzhiivanov, Nedyalko Nenov Communicated by V. Drensky Abstract. Let Γ(M) where M V (G) be the set of all vertices

### The Topology of Spaces of Triads

The Topology of Spaces of Triads Michael J. Catanzaro August 28, 2009 Abstract Major and minor triads in the diatonic scale can be naturally assembled into a space which turns out to be a torus. We study

### TOURING PROBLEMS A MATHEMATICAL APPROACH

TOURING PROBLEMS A MATHEMATICAL APPROACH VIPUL NAIK Abstract. The article surveys touring, a problem of fun math, and employs serious techniques to attack it. The text is suitable for high school students

### Triangle deletion. Ernie Croot. February 3, 2010

Triangle deletion Ernie Croot February 3, 2010 1 Introduction The purpose of this note is to give an intuitive outline of the triangle deletion theorem of Ruzsa and Szemerédi, which says that if G = (V,

### 136 CHAPTER 4. INDUCTION, GRAPHS AND TREES

136 TER 4. INDUCTION, GRHS ND TREES 4.3 Graphs In this chapter we introduce a fundamental structural idea of discrete mathematics, that of a graph. Many situations in the applications of discrete mathematics

### Class One: Degree Sequences

Class One: Degree Sequences For our purposes a graph is a just a bunch of points, called vertices, together with lines or curves, called edges, joining certain pairs of vertices. Three small examples of

### L(n) OA(n, 3): if L is a latin square with colour set [n] then the following is an O(n, 3):

9 Latin Squares Latin Square: A Latin Square of order n is an n n array where each cell is occupied by one of n colours with the property that each colour appears exactly once in each row and in each column.

### Finding the Shortest Move-Sequence in the Graph-Generalized 15-Puzzle is NP-Hard

Finding the Shortest Move-Sequence in the Graph-Generalized 15-Puzzle is NP-Hard Oded Goldreich Abstract. Following Wilson (J. Comb. Th. (B), 1975), Johnson (J. of Alg., 1983), and Kornhauser, Miller and

### Tilings of the sphere with right triangles III: the asymptotically obtuse families

Tilings of the sphere with right triangles III: the asymptotically obtuse families Robert J. MacG. Dawson Department of Mathematics and Computing Science Saint Mary s University Halifax, Nova Scotia, Canada

### Counting unlabelled topologies and transitive relations

Counting unlabelled topologies and transitive relations Gunnar Brinkmann Applied Mathematics and Computer Science Ghent University B 9000 Ghent, Belgium Gunnar.Brinkmann@ugent.be Brendan D. McKay Department

### Chapter 7: Products and quotients

Chapter 7: Products and quotients Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 42, Spring 24 M. Macauley (Clemson) Chapter 7: Products

GROUP ACTIONS KEITH CONRAD 1. Introduction The symmetric groups S n, alternating groups A n, and (for n 3) dihedral groups D n behave, by their very definition, as permutations on certain sets. The groups