Ringing the Changes II
|
|
- Lenard Watts
- 5 years ago
- Views:
Transcription
1 Ringing the Changes II
2 Cayley Color Graphs Let G be a group and H a subset of G. We define a Cayley digraph (directed graph) by: Each element of G is a vertex of the digraph Γ and an arc is drawn from vertex a to vertex b if there is an element h H so that b = ha in G. If H = H -1 (the inverse of every element in H is also in H), then for every arc from a to b there is also an arc from b to a. In this situation it is common to replace the pair of arcs by an undirected edge and the Cayley digraph is called a Cayley graph. When each element of H is its own inverse, we may label the edges of the Cayley graph by the elements of H which determine the edge. This labeled graph is called a Cayley color graph.
3 Cayley Color Graphs Let G be the symmetric group S 3 whose elements are (written in cyclic notation): G = {I, (12), (13), (23), (123), (132)}. Let H = {(12), (132)}. The Cayley digraph C G (H) is: I (12) (132) (13) = (132)(12) (23) = (12)(132) (123)
4 Cayley Color Graphs Let H' = {(123), (132)}. The Cayley graph C G (H') is: I (12) (132) (13) (23) (123)
5 Cayley Color Graphs Let H'' = {(12), (13)}. The Cayley color graph C G (H'') is: I (12) (123) (23) (12) (13) (13) (132)
6 Ringing the Changes We can use Cayley color graphs to model change ringing. For n bells, we will use the symmetric group S n which consists of all the permutations on n symbols. The transitions (permutations which either fix or interchange adjacent bells) form the set H. Since each transition is an involution (has order 2), we can form the Cayley color graph C Sn (H). To obtain an extent on n bells which satisfies rules 1, 2 and 3 we need only find a Hamiltonian cycle (which visits each vertex exactly once before returning to the starting vertex) in this color graph and starting at the identity vertex, trace the cycle using the edge labels to obtain each change.
7 Hamiltonian Cycles As an example consider an extent of 3-bells (singles). The Cayley color graph is on S 3 and H = {(12),(23)}. I (12) (123) (23) (12) = a (23) = b (13) (132) Hamiltonian: ababab = (ab) 3 quick six bababa = (ba) 3 slow six Quick Six 123 = I 213 = (12) 231 = (123) 321 = (13) 312 = (132) 132 = (23) 123 = I
8 Another Example Consider the Cayley color graph of S 4 with H = {(12), (23), (34)}. Starting with (12) this is Double Court Minimus.
9 Another Example Consider the Cayley color graph of S 4 with H = {(12), (23), (34)}. Starting at JT gives the Johnson-Trotter sequence. JT Starting with (12) this is Double Canterbury Minimus.
10 Finding Hamiltonian Cycles Graphs containing a Hamiltonian cycle are called Hamiltonian graphs. Deciding whether or not a graph is Hamiltonian is an NP-problem, meaning that there is no known fast algorithm that will answer this question for all graphs. There are some theorems which give sufficient (but not necessary) conditions for a graph to be Hamiltonian (Ore, Dirac, Chvátal) but none of these are applicable in our situation. For small graphs we can use a basic backtracking algorithm to find Hamiltonian cycles, but in our case there is a theoretical result which is useful.
11 Rapaport's Construction In Cayley colour groups and Hamilton lines [Scripta Mathematica 24(1959), pp.51-58], Elvira Strasser Rapaport proved that the Cayley color graphs we are interested in are in fact Hamiltonian. Her proof is based on the following lemma: Lemma: A connected graph, regular of degree three, has a Hamiltonian cycle if there exists a set P of cycles and a set Q of 4-cycles each partitioning the vertex set of the graph and such that no member of P contains every vertex of a member of Q.
12 Rapaport's Lemma Lemma: A connected graph, regular of degree three, has a Hamiltonian cycle if there exists a set P of cycles and a set Q of 4-cycles each partitioning the vertex set of the graph and such that no member of P contains every vertex of a member of Q. Proof: Repeat this operation until it can't be done again: P 1 Q 1 P 2 P 1, P 2 P, Q 1 Q P has become P', still a disjoint union of cycles covering the vertex set. If P' is not a single cycle, it contains two cycles C 1 and C 2 joined by an edge E = v 1 v 2, v 1 C 1, v 2 C 2 (since the graph is connected).
13 Rapaport's Lemma Lemma: A connected graph, regular of degree three, has a Hamiltonian cycle if there exists a set P of cycles and a set Q of 4-cycles each partitioning the vertex set of the graph and such that no member of P contains every vertex of a member of Q. Proof (cont.): If E is in some Q i, then the two other edges through v 1 must be in C 1 and one of them, say v 1 p 1 is in Q i. Similarly, an edge v 2 p 2 in C 2 is also in Q i. We could thus carry out the above operation again, a contradiction. Thus, E is in no Q i. The edges v 1 p 1 and v 1 p 3 are in C 1. As v 1 is in some Q 1, these two edges must also be in Q 1 (as E is not and valency is 3). C 1 thus contains 3 vertices and 2 consecutive edges of Q 1. The consecutive edges could not have arisen from the operation, so these edges must have been in some P i to start with. The 4 th vertex of Q 1 could not be in this P i by hypothesis.
14 Rapaport's Lemma Lemma: A connected graph, regular of degree three, has a Hamiltonian cycle if there exists a set P of cycles and a set Q of 4-cycles each partitioning the vertex set of the graph and such that no member of P contains every vertex of a member of Q. Proof (cont.): This 4 th vertex of Q 1, say V, can not be on any of the P j in P. If it were, it would have to be connected to two vertices of its cycle P j and also two vertices of cycle P i, but valency 3 would then force these two cycles to have a common vertex, a contradiction.
15 Rapaport's Theorem Theorem: For n 3, the Cayley color graph C sn (H) is Hamiltonian when H = {g 1 =(12), g 2 =(12)(34)(56)..., g 3 = (23)(45)(67)... }. Pf: The permutations (12) and ( ) {= g 2 g 3 } are known to generate S n, so the three permutations of H certainly do. This means that the Cayley color graph is connected. For n > 4, start at any vertex and follow the path g 1 g 2 g 1 g 2. Since this product is the identity transformation, the path determines a 4-cycle. If two of the 4-cycles generated this way had a vertex in common, they would have to be identical. Let Q be the set of all such 4- cycles. There are n!/4 such 4-cycles in Q.
16 Rapaport's Theorem Theorem: For n 3, the Cayley color graph C sn (H) is Hamiltonian when H = {g 1 =(12), g 2 =(12)(34)(56)..., g 3 = (23)(45)(67)... }. Pf(cont): The set P consists of the 12-cycles that are formed from the relation (g 1 g 3 ) 6 = id. There are n!/12 disjoint 12-cycles in P. As no product of alternating g 3 and g 1 can equal g 2 (consider what happens to 4 and 5) none of these 12-cycles can contain all (or even 3) vertices of one of Q's 4-cycles. Thus, the conditions of the lemma are satisfied and a Hamiltonian cycle exists in this graph. If n = 3, H = {g 1 = g 2 = (12), g 3 = (23)} and the quick six singles gives the result. If n = 4, replace the 12- cycles of P by the 6-cycles generated by (g 1 g 3 ) 3 =id.
17 Example when n = 4 α I β (12) (12)(34) (23) η λ η λ β α
18 Other Choices Although Rapaport did not use them, P could also be made up of 2n-cycles which come from (g 2 g 3 ) n = id. There are n!/2n = ½(n-1)! of these 2n-cycles. For n = 4 we would have 3 8-cycles forming P, and the algorithm of the lemma will still work.
19 Example when n = 4 α I β (12) (12)(34) (23) η λ η λ β α
20 Other Choices However, as the following examples show, not all Hamiltonian cycles come from this construction of Rapaport.
21 Example when n = 4 α I β (12) (12)(34) (23) η λ η λ β α
22 Example when n = 4 α I β (12) (12)(34) (23) η λ η λ Single Court starting with (12) β α
23 Example when n = 4 α I β (12) (12)(34) (23) η λ η λ β α Reverse Bob starts with (12)(34)
24 Euler Characteristic The Cayley graphs we have examined so far are not always planar graphs. We can embed a graph that we are interested in in a planar way (no edges crossing except at vertices) on a surface with a certain number of holes. If a connected graph with n vertices and e edges has a planar embedding in a plane or on a sphere then n e + f = 2 where f is the number of faces (minimal [having no chords] cycles) of the graph. Example: n = 4, e = 6, f = 4 (don't forget the outside face) = 2
25 Euler Characteristic The quantity n e + f is called the Euler characteristic of the graph and is related to the type of surface in which the graph may have a planar embedding. The result here is: Theorem: Suppose that a graph with n vertices, e edges and f faces has a simple imbedding in a closed surface S, then (1) If S is a sphere with g handles then n-e+f = 2 2g (2) If S is a sphere with c cross-caps then n-e+f = 2 - c. A simple imbedding is one in which each face (including the infinite face) is topologically a disk. The surfaces of type (1) are orientable while those of type (2) are nonorientable.
26 Embedding Graphs Sphere with 1 handle = Torus (Doughnut)
27 Embedding Graphs Torus Möbius Strip
28 Projective Plane Embedding Graphs Cross Cap
29 Embedding Rapaport's Graph The Rapaport graph has n! vertices, 3n!/2 edges and its faces are the n!/4 4-cycles, n!/12 12-cycles and ½(n-1)! 2n-cycles for n > 4. Thus the Euler characteristic is n! - 3n!/2 + (n!/4 + n!/12 + n!/2n) = n! ( 1 3/2 + 1/4 + 1/12 + 1/2n) = n!(-1/6 + 1/2n) n! (3 n)/6n = (n-1)! (3 n)/6 = 2 [(n-1)(n-2)(n-3) 2 (n-4)! + 12]/6. For n = 4 we have 24 vertices, 36 edges, faces, so n e + f = = 1 = 2 1 and so, it may be embedded in a projective plane.
Campanology - Ringing the changes
Department of Mathematics Bachelor Thesis Campanology - Ringing the changes Fabia Weber supervised by PD Dr. Lorenz Halbeisen August 17, 2017 Contents Introduction...............................................
More informationarxiv: v2 [math.gr] 17 Dec 2017
The complement of proper power graphs of finite groups T. Anitha, R. Rajkumar arxiv:1601.03683v2 [math.gr] 17 Dec 2017 Department of Mathematics, The Gandhigram Rural Institute Deemed to be University,
More informationPlanar Graphs (1) Planar Graphs (3) Planar Graphs (2) Planar Graphs (4)
S-72.2420/T-79.5203 Planarity; Edges and Cycles 1 Planar Graphs (1) Topological graph theory, broadly conceived, is the study of graph layouts. Contemporary applications include circuit layouts on silicon
More informationUNIQUENESS OF HIGHLY REPRESENTATIVE SURFACE EMBEDDINGS
UNIQUENESS OF HIGHLY REPRESENTATIVE SURFACE EMBEDDINGS P. D. Seymour Bellcore 445 South St. Morristown, New Jersey 07960, USA and Robin Thomas 1 School of Mathematics Georgia Institute of Technology Atlanta,
More informationON HAMILTON CIRCUITS IN CAYLEY DIGRAPHS OVER GENERALIZED DIHEDRAL GROUPS
REVISTA DE LA UNIÓN MATEMÁTICA ARGENTINA Vol. 53, No. 2, 2012, 79 87 ON HAMILTON CIRCUITS IN CAYLEY DIGRAPHS OVER GENERALIZED DIHEDRAL GROUPS ADRIÁN PASTINE AND DANIEL JAUME Abstract. In this paper we
More informationHAMILTONICITY IN CAYLEY GRAPHS AND DIGRAPHS OF FINITE ABELIAN GROUPS.
HAMILTONICITY IN CAYLEY GRAPHS AND DIGRAPHS OF FINITE ABELIAN GROUPS. MARY STELOW Abstract. Cayley graphs and digraphs are introduced, and their importance and utility in group theory is formally shown.
More informationHamiltonicity of digraphs for universal cycles of permutations
Hamiltonicity of digraphs for universal cycles of permutations Garth Isaak Abstract The digraphs P (n, k) have vertices corresponding to length k permutations of an n set and arcs corresponding to (k +
More informationThree-coloring triangle-free graphs on surfaces VII. A linear-time algorithm
Three-coloring triangle-free graphs on surfaces VII. A linear-time algorithm Zdeněk Dvořák Daniel Král Robin Thomas Abstract We give a linear-time algorithm to decide 3-colorability of a trianglefree graph
More informationTOWARDS A SPLITTER THEOREM FOR INTERNALLY 4-CONNECTED BINARY MATROIDS VII
TOWARDS A SPLITTER THEOREM FOR INTERNALLY 4-CONNECTED BINARY MATROIDS VII CAROLYN CHUN AND JAMES OXLEY Abstract. Let M be a 3-connected binary matroid; M is internally 4- connected if one side of every
More informationARRANGEABILITY AND CLIQUE SUBDIVISIONS. Department of Mathematics and Computer Science Emory University Atlanta, GA and
ARRANGEABILITY AND CLIQUE SUBDIVISIONS Vojtěch Rödl* Department of Mathematics and Computer Science Emory University Atlanta, GA 30322 and Robin Thomas** School of Mathematics Georgia Institute of Technology
More informationFine structure of 4-critical triangle-free graphs III. General surfaces
Fine structure of 4-critical triangle-free graphs III. General surfaces Zdeněk Dvořák Bernard Lidický February 16, 2017 Abstract Dvořák, Král and Thomas [4, 6] gave a description of the structure of triangle-free
More informationDirac s Map-Color Theorem for Choosability
Dirac s Map-Color Theorem for Choosability T. Böhme B. Mohar Technical University of Ilmenau, University of Ljubljana, D-98684 Ilmenau, Germany Jadranska 19, 1111 Ljubljana, Slovenia M. Stiebitz Technical
More informationarxiv: v1 [cs.ds] 2 Oct 2018
Contracting to a Longest Path in H-Free Graphs Walter Kern 1 and Daniël Paulusma 2 1 Department of Applied Mathematics, University of Twente, The Netherlands w.kern@twente.nl 2 Department of Computer Science,
More information8.5 Sequencing Problems
8.5 Sequencing Problems Basic genres. Packing problems: SET-PACKING, INDEPENDENT SET. Covering problems: SET-COVER, VERTEX-COVER. Constraint satisfaction problems: SAT, 3-SAT. Sequencing problems: HAMILTONIAN-CYCLE,
More informationImprimitive symmetric graphs with cyclic blocks
Imprimitive symmetric graphs with cyclic blocks Sanming Zhou Department of Mathematics and Statistics University of Melbourne Joint work with Cai Heng Li and Cheryl E. Praeger December 17, 2008 Outline
More informationarxiv: v1 [cs.dm] 12 Jun 2016
A Simple Extension of Dirac s Theorem on Hamiltonicity Yasemin Büyükçolak a,, Didem Gözüpek b, Sibel Özkana, Mordechai Shalom c,d,1 a Department of Mathematics, Gebze Technical University, Kocaeli, Turkey
More informationResearch Article Nonnormal Edge-Transitive Cubic Cayley Graphs of Dihedral Groups
International Scholarly Research Network ISRN Algebra Volume 2011, Article ID 428959, 6 pages doi:10.5402/2011/428959 Research Article Nonnormal Edge-Transitive Cubic Cayley Graphs of Dihedral Groups Mehdi
More informationFine Structure of 4-Critical Triangle-Free Graphs II. Planar Triangle-Free Graphs with Two Precolored 4-Cycles
Mathematics Publications Mathematics 2017 Fine Structure of 4-Critical Triangle-Free Graphs II. Planar Triangle-Free Graphs with Two Precolored 4-Cycles Zdeněk Dvořák Charles University Bernard Lidicky
More informationHamiltonian circuits in Cayley digraphs. Dan Isaksen. Wayne State University
Hamiltonian circuits in Cayley digraphs Dan Isaksen Wayne State University 1 Digraphs Definition. A digraph is a set V and a subset E of V V. The elements of V are called vertices. We think of vertices
More informationA well-quasi-order for tournaments
A well-quasi-order for tournaments Maria Chudnovsky 1 Columbia University, New York, NY 10027 Paul Seymour 2 Princeton University, Princeton, NJ 08544 June 12, 2009; revised April 19, 2011 1 Supported
More informationCS 301: Complexity of Algorithms (Term I 2008) Alex Tiskin Harald Räcke. Hamiltonian Cycle. 8.5 Sequencing Problems. Directed Hamiltonian Cycle
8.5 Sequencing Problems Basic genres. Packing problems: SET-PACKING, INDEPENDENT SET. Covering problems: SET-COVER, VERTEX-COVER. Constraint satisfaction problems: SAT, 3-SAT. Sequencing problems: HAMILTONIAN-CYCLE,
More informationPaul Erdős and Graph Ramsey Theory
Paul Erdős and Graph Ramsey Theory Benny Sudakov ETH and UCLA Ramsey theorem Ramsey theorem Definition: The Ramsey number r(s, n) is the minimum N such that every red-blue coloring of the edges of a complete
More informationOdd Components of Co-Trees and Graph Embeddings 1
Odd Components of Co-Trees and Graph Embeddings 1 Han Ren, Dengju Ma and Junjie Lu Department of Mathematics, East China Normal University, Shanghai 0006, P.C.China Abstract: In this paper we investigate
More informationLaplacian Integral Graphs with Maximum Degree 3
Laplacian Integral Graphs with Maximum Degree Steve Kirkland Department of Mathematics and Statistics University of Regina Regina, Saskatchewan, Canada S4S 0A kirkland@math.uregina.ca Submitted: Nov 5,
More informationarxiv: v2 [math.co] 7 Jan 2016
Global Cycle Properties in Locally Isometric Graphs arxiv:1506.03310v2 [math.co] 7 Jan 2016 Adam Borchert, Skylar Nicol, Ortrud R. Oellermann Department of Mathematics and Statistics University of Winnipeg,
More informationPolynomial-time Reductions
Polynomial-time Reductions Disclaimer: Many denitions in these slides should be taken as the intuitive meaning, as the precise meaning of some of the terms are hard to pin down without introducing the
More informationMassachusetts Institute of Technology 6.042J/18.062J, Fall 02: Mathematics for Computer Science Professor Albert Meyer and Dr.
Massachusetts Institute of Technology 6.042J/18.062J, Fall 02: Mathematics for Computer Science Professor Albert Meyer and Dr. Radhika Nagpal Quiz 1 Appendix Appendix Contents 1 Induction 2 2 Relations
More information1 Spaces and operations Continuity and metric spaces Topological spaces Compactness... 3
Compact course notes Topology I Fall 2011 Professor: A. Penskoi transcribed by: J. Lazovskis Independent University of Moscow December 23, 2011 Contents 1 Spaces and operations 2 1.1 Continuity and metric
More informationHamiltonian Graphs Graphs
COMP2121 Discrete Mathematics Hamiltonian Graphs Graphs Hubert Chan (Chapter 9.5) [O1 Abstract Concepts] [O2 Proof Techniques] [O3 Basic Analysis Techniques] 1 Hamiltonian Paths and Circuits [O1] A Hamiltonian
More informationSome results on the reduced power graph of a group
Some results on the reduced power graph of a group R. Rajkumar and T. Anitha arxiv:1804.00728v1 [math.gr] 2 Apr 2018 Department of Mathematics, The Gandhigram Rural Institute-Deemed to be University, Gandhigram
More informationGraphs and digraphs with all 2 factors isomorphic
Graphs and digraphs with all 2 factors isomorphic M. Abreu, Dipartimento di Matematica, Università della Basilicata, I-85100 Potenza, Italy. e-mail: abreu@math.fau.edu R.E.L. Aldred, University of Otago,
More informationPlanar Ramsey Numbers for Small Graphs
Planar Ramsey Numbers for Small Graphs Andrzej Dudek Department of Mathematics and Computer Science Emory University Atlanta, GA 30322, USA Andrzej Ruciński Faculty of Mathematics and Computer Science
More informationThe power graph of a finite group, II
The power graph of a finite group, II Peter J. Cameron School of Mathematical Sciences Queen Mary, University of London Mile End Road London E1 4NS, U.K. Abstract The directed power graph of a group G
More informationThree-coloring triangle-free graphs on surfaces III. Graphs of girth five
Three-coloring triangle-free graphs on surfaces III. Graphs of girth five Zdeněk Dvořák Daniel Král Robin Thomas Abstract We show that the size of a 4-critical graph of girth at least five is bounded by
More informationHAMILTON CYCLES IN CAYLEY GRAPHS
Hamiltonicity of (2, s, 3)- University of Primorska July, 2011 Hamiltonicity of (2, s, 3)- Lovász, 1969 Does every connected vertex-transitive graph have a Hamilton path? Hamiltonicity of (2, s, 3)- Hamiltonicity
More informationWe would like a theorem that says A graph G is hamiltonian if and only if G has property Q, where Q can be checked in polynomial time.
9 Tough Graphs and Hamilton Cycles We would like a theorem that says A graph G is hamiltonian if and only if G has property Q, where Q can be checked in polynomial time. However in the early 1970 s it
More informationThree-coloring triangle-free graphs on surfaces IV. Bounding face sizes of 4-critical graphs
Three-coloring triangle-free graphs on surfaces IV. Bounding face sizes of 4-critical graphs Zdeněk Dvořák Daniel Král Robin Thomas Abstract Let G be a 4-critical graph with t triangles, embedded in a
More informationChapter 34: NP-Completeness
Graph Algorithms - Spring 2011 Set 17. Lecturer: Huilan Chang Reference: Cormen, Leiserson, Rivest, and Stein, Introduction to Algorithms, 2nd Edition, The MIT Press. Chapter 34: NP-Completeness 2. Polynomial-time
More information3-coloring triangle-free planar graphs with a precolored 8-cycle
-coloring triangle-free planar graphs with a precolored 8-cycle Zdeněk Dvořák Bernard Lidický May 0, 0 Abstract Let G be a planar triangle-free graph and led C be a cycle in G of length at most 8. We characterize
More informationSelected Topics in AGT Lecture 4 Introduction to Schur Rings
Selected Topics in AGT Lecture 4 Introduction to Schur Rings Mikhail Klin (BGU and UMB) September 14 18, 2015 M. Klin Selected topics in AGT September 2015 1 / 75 1 Schur rings as a particular case of
More informationA Family of One-regular Graphs of Valency 4
Europ. J. Combinatorics (1997) 18, 59 64 A Family of One-regular Graphs of Valency 4 D RAGAN M ARUSä ICä A graph is said to be one - regular if its automorphism group acts regularly on the set of its arcs.
More informationGraphs with large maximum degree containing no odd cycles of a given length
Graphs with large maximum degree containing no odd cycles of a given length Paul Balister Béla Bollobás Oliver Riordan Richard H. Schelp October 7, 2002 Abstract Let us write f(n, ; C 2k+1 ) for the maximal
More informationApplications of the Lopsided Lovász Local Lemma Regarding Hypergraphs
Regarding Hypergraphs Ph.D. Dissertation Defense April 15, 2013 Overview The Local Lemmata 2-Coloring Hypergraphs with the Original Local Lemma Counting Derangements with the Lopsided Local Lemma Lopsided
More informationP versus NP. Math 40210, Fall November 10, Math (Fall 2015) P versus NP November 10, / 9
P versus NP Math 40210, Fall 2015 November 10, 2015 Math 40210 (Fall 2015) P versus NP November 10, 2015 1 / 9 Properties of graphs A property of a graph is anything that can be described without referring
More informationPhysical Mapping. Restriction Mapping. Lecture 12. A physical map of a DNA tells the location of precisely defined sequences along the molecule.
Computat onal iology Lecture 12 Physical Mapping physical map of a DN tells the location of precisely defined sequences along the molecule. Restriction mapping: mapping of restriction sites of a cutting
More information6 Euler Circuits and Hamiltonian Cycles
November 14, 2017 6 Euler Circuits and Hamiltonian Cycles William T. Trotter trotter@math.gatech.edu EulerTrails and Circuits Definition A trail (x 1, x 2, x 3,, x t) in a graph G is called an Euler trail
More informationCSCE 551 Final Exam, April 28, 2016 Answer Key
CSCE 551 Final Exam, April 28, 2016 Answer Key 1. (15 points) Fix any alphabet Σ containing the symbol a. For any language L Σ, define the language a\l := {w Σ wa L}. Show that if L is regular, then a\l
More information8. INTRACTABILITY I. Lecture slides by Kevin Wayne Copyright 2005 Pearson-Addison Wesley. Last updated on 2/6/18 2:16 AM
8. INTRACTABILITY I poly-time reductions packing and covering problems constraint satisfaction problems sequencing problems partitioning problems graph coloring numerical problems Lecture slides by Kevin
More informationOn non-hamiltonian circulant digraphs of outdegree three
On non-hamiltonian circulant digraphs of outdegree three Stephen C. Locke DEPARTMENT OF MATHEMATICAL SCIENCES, FLORIDA ATLANTIC UNIVERSITY, BOCA RATON, FL 33431 Dave Witte DEPARTMENT OF MATHEMATICS, OKLAHOMA
More information5. Partitions and Relations Ch.22 of PJE.
5. Partitions and Relations Ch. of PJE. We now generalize the ideas of congruence classes of Z to classes of any set X. The properties of congruence classes that we start with here are that they are disjoint
More informationSergey Norin Department of Mathematics and Statistics McGill University Montreal, Quebec H3A 2K6, Canada. and
NON-PLANAR EXTENSIONS OF SUBDIVISIONS OF PLANAR GRAPHS Sergey Norin Department of Mathematics and Statistics McGill University Montreal, Quebec H3A 2K6, Canada and Robin Thomas 1 School of Mathematics
More informationConstructing Linkages for Drawing Plane Curves
Constructing Linkages for Drawing Plane Curves Christoph Koutschan (joint work with Matteo Gallet, Zijia Li, Georg Regensburger, Josef Schicho, Nelly Villamizar) Johann Radon Institute for Computational
More informationBasic definitions. Remarks
Basic definitions In a graph G(V, E) a Hamiltonian tour sometimes also called a Hamiltonian cycle is a closed trail that includes each of the graph s vertices exactly once. A graph that contains such a
More information4/12/2011. Chapter 8. NP and Computational Intractability. Directed Hamiltonian Cycle. Traveling Salesman Problem. Directed Hamiltonian Cycle
Directed Hamiltonian Cycle Chapter 8 NP and Computational Intractability Claim. G has a Hamiltonian cycle iff G' does. Pf. Suppose G has a directed Hamiltonian cycle Γ. Then G' has an undirected Hamiltonian
More informationP P P NP-Hard: L is NP-hard if for all L NP, L L. Thus, if we could solve L in polynomial. Cook's Theorem and Reductions
Summary of the previous lecture Recall that we mentioned the following topics: P: is the set of decision problems (or languages) that are solvable in polynomial time. NP: is the set of decision problems
More informationUNAVOIDABLE INDUCED SUBGRAPHS IN LARGE GRAPHS WITH NO HOMOGENEOUS SETS
UNAVOIDABLE INDUCED SUBGRAPHS IN LARGE GRAPHS WITH NO HOMOGENEOUS SETS MARIA CHUDNOVSKY, RINGI KIM, SANG-IL OUM, AND PAUL SEYMOUR Abstract. An n-vertex graph is prime if it has no homogeneous set, that
More informationParity Versions of 2-Connectedness
Parity Versions of 2-Connectedness C. Little Institute of Fundamental Sciences Massey University Palmerston North, New Zealand c.little@massey.ac.nz A. Vince Department of Mathematics University of Florida
More informationComplete graphs whose topological symmetry groups are polyhedral
Digital Commons@ Loyola Marymount University and Loyola Law School Mathematics Faculty Works Mathematics 1-1-2011 Complete graphs whose topological symmetry groups are polyhedral Eric Flapan Blake Mellor
More informationData Structures and Algorithms
Data Structures and Algorithms CS245-2015S-23 NP-Completeness and Undecidablity David Galles Department of Computer Science University of San Francisco 23-0: Hard Problems Some algorithms take exponential
More informationSUB-EXPONENTIALLY MANY 3-COLORINGS OF TRIANGLE-FREE PLANAR GRAPHS
SUB-EXPONENTIALLY MANY 3-COLORINGS OF TRIANGLE-FREE PLANAR GRAPHS Arash Asadi Luke Postle 1 Robin Thomas 2 School of Mathematics Georgia Institute of Technology Atlanta, Georgia 30332-0160, USA ABSTRACT
More informationContact author address Dragan Marusic 3 IMFM, Oddelek za matematiko Univerza v Ljubljani Jadranska 19, 1111 Ljubljana Slovenija Tel.: F
PERMUTATION GROUPS, VERTEX-TRANSITIVE DIGRAPHS AND SEMIREGULAR AUTOMORPHISMS Dragan Marusic 1 Raffaele Scapellato 2 IMFM, Oddelek za matematiko Univerza v Ljubljani Jadranska 19, 61111 Ljubljana Slovenija
More informationP versus NP. Math 40210, Spring September 16, Math (Spring 2012) P versus NP September 16, / 9
P versus NP Math 40210, Spring 2012 September 16, 2012 Math 40210 (Spring 2012) P versus NP September 16, 2012 1 / 9 Properties of graphs A property of a graph is anything that can be described without
More informationThe Manhattan Product of Digraphs
Electronic Journal of Graph Theory and Applications 1 (1 (2013, 11 27 The Manhattan Product of Digraphs F. Comellas, C. Dalfó, M.A. Fiol Departament de Matemàtica Aplicada IV, Universitat Politècnica de
More information4 CONNECTED PROJECTIVE-PLANAR GRAPHS ARE HAMILTONIAN. Robin Thomas* Xingxing Yu**
4 CONNECTED PROJECTIVE-PLANAR GRAPHS ARE HAMILTONIAN Robin Thomas* Xingxing Yu** School of Mathematics Georgia Institute of Technology Atlanta, Georgia 30332, USA May 1991, revised 23 October 1993. Published
More informationLatin squares: Equivalents and equivalence
Latin squares: Equivalents and equivalence 1 Introduction This essay describes some mathematical structures equivalent to Latin squares and some notions of equivalence of such structures. According to
More informationChapter 8. NP and Computational Intractability. Slides by Kevin Wayne. Copyright 2005 Pearson-Addison Wesley. All rights reserved.
Chapter 8 NP and Computational Intractability Slides by Kevin Wayne. Copyright 2005 Pearson-Addison Wesley. All rights reserved. 1 8.5 Sequencing Problems Basic genres.! Packing problems: SET-PACKING,
More information1 More finite deterministic automata
CS 125 Section #6 Finite automata October 18, 2016 1 More finite deterministic automata Exercise. Consider the following game with two players: Repeatedly flip a coin. On heads, player 1 gets a point.
More informationAlgorithms and Theory of Computation. Lecture 22: NP-Completeness (2)
Algorithms and Theory of Computation Lecture 22: NP-Completeness (2) Xiaohui Bei MAS 714 November 8, 2018 Nanyang Technological University MAS 714 November 8, 2018 1 / 20 Set Cover Set Cover Input: a set
More information4. How to prove a problem is NPC
The reducibility relation T is transitive, i.e, A T B and B T C imply A T C Therefore, to prove that a problem A is NPC: (1) show that A NP (2) choose some known NPC problem B define a polynomial transformation
More informationCombining the cycle index and the Tutte polynomial?
Combining the cycle index and the Tutte polynomial? Peter J. Cameron University of St Andrews Combinatorics Seminar University of Vienna 23 March 2017 Selections Students often meet the following table
More informationClaw-free Graphs. III. Sparse decomposition
Claw-free Graphs. III. Sparse decomposition Maria Chudnovsky 1 and Paul Seymour Princeton University, Princeton NJ 08544 October 14, 003; revised May 8, 004 1 This research was conducted while the author
More informationMATH 22 HAMILTONIAN GRAPHS. Lecture V: 11/18/2003
MATH 22 Lecture V: 11/18/2003 HAMILTONIAN GRAPHS All communities [graphs] divide themselves into the few and the many [i.e., are bipartite]. Alexander Hamilton, Debates of the Federal Convention Before
More informationOn cyclic decompositions of K n+1,n+1 I into a 2-regular graph with at most 2 components
On cyclic decompositions of K n+1,n+1 I into a 2-regular graph with at most 2 components R. C. Bunge 1, S. I. El-Zanati 1, D. Gibson 2, U. Jongthawonwuth 3, J. Nagel 4, B. Stanley 5, A. Zale 1 1 Department
More informationMalaya J. Mat. 2(3)(2014)
Malaya J Mat (3)(04) 80-87 On k-step Hamiltonian Bipartite and Tripartite Graphs Gee-Choon Lau a,, Sin-Min Lee b, Karl Schaffer c and Siu-Ming Tong d a Faculty of Comp & Mathematical Sciences, Universiti
More informationOrdering and Reordering: Using Heffter Arrays to Biembed Complete Graphs
University of Vermont ScholarWorks @ UVM Graduate College Dissertations and Theses Dissertations and Theses 2015 Ordering and Reordering: Using Heffter Arrays to Biembed Complete Graphs Amelia Mattern
More informationarxiv: v1 [cs.ds] 20 Feb 2017
AN OPTIMAL XP ALGORITHM FOR HAMILTONIAN CYCLE ON GRAPHS OF BOUNDED CLIQUE-WIDTH BENJAMIN BERGOUGNOUX, MAMADOU MOUSTAPHA KANTÉ, AND O-JOUNG KWON arxiv:1702.06095v1 [cs.ds] 20 Feb 2017 Abstract. For MSO
More informationFine structure of 4-critical triangle-free graphs III. General surfaces
Mathematics Publications Mathematics 2018 Fine structure of 4-critical triangle-free graphs III. General surfaces Zdenek Dvorak Charles University, Prague Bernard Lidicky Iowa State University, lidicky@iastate.edu
More informationarxiv: v3 [cs.dm] 18 Oct 2017
Decycling a Graph by the Removal of a Matching: Characterizations for Special Classes arxiv:1707.02473v3 [cs.dm] 18 Oct 2017 Fábio Protti and Uéverton dos Santos Souza Institute of Computing - Universidade
More informationNP and Computational Intractability
NP and Computational Intractability 1 Review Basic reduction strategies. Simple equivalence: INDEPENDENT-SET P VERTEX-COVER. Special case to general case: VERTEX-COVER P SET-COVER. Encoding with gadgets:
More informationarxiv:quant-ph/ v1 15 Apr 2005
Quantum walks on directed graphs Ashley Montanaro arxiv:quant-ph/0504116v1 15 Apr 2005 February 1, 2008 Abstract We consider the definition of quantum walks on directed graphs. Call a directed graph reversible
More informationDistance between two k-sets and Path-Systems Extendibility
Distance between two k-sets and Path-Systems Extendibility December 2, 2003 Ronald J. Gould (Emory University), Thor C. Whalen (Metron, Inc.) Abstract Given a simple graph G on n vertices, let σ 2 (G)
More informationTheory of Computation Chapter 9
0-0 Theory of Computation Chapter 9 Guan-Shieng Huang May 12, 2003 NP-completeness Problems NP: the class of languages decided by nondeterministic Turing machine in polynomial time NP-completeness: Cook
More informationA dynamic programming approach to generation of strong traces
A dynamic programming approach to generation of strong traces Nino Bašić Faculty of Mathematics, Natural Sciences and Information Technologies University of Primorska 32 nd TBI Winterseminar Bled, 16 February
More informationPaths and cycles in extended and decomposable digraphs
Paths and cycles in extended and decomposable digraphs Jørgen Bang-Jensen Gregory Gutin Department of Mathematics and Computer Science Odense University, Denmark Abstract We consider digraphs called extended
More informationHow many randomly colored edges make a randomly colored dense graph rainbow hamiltonian or rainbow connected?
How many randomly colored edges make a randomly colored dense graph rainbow hamiltonian or rainbow connected? Michael Anastos and Alan Frieze February 1, 2018 Abstract In this paper we study the randomly
More informationFolding graphs and applications, d après Stallings
Folding graphs and applications, d après Stallings Mladen Bestvina Fall 2001 Class Notes, updated 2010 1 Folding and applications A graph is a 1-dimensional cell complex. Thus we can have more than one
More information5 Flows and cuts in digraphs
5 Flows and cuts in digraphs Recall that a digraph or network is a pair G = (V, E) where V is a set and E is a multiset of ordered pairs of elements of V, which we refer to as arcs. Note that two vertices
More informationMcGill University Faculty of Science. Solutions to Practice Final Examination Math 240 Discrete Structures 1. Time: 3 hours Marked out of 60
McGill University Faculty of Science Solutions to Practice Final Examination Math 40 Discrete Structures Time: hours Marked out of 60 Question. [6] Prove that the statement (p q) (q r) (p r) is a contradiction
More informationFor a group G and a subgroup H G, denoteby[g:h] the set of right cosets of H in G, thatis
Chapter 3 Coset Graphs We have seen that every transitive action is equivalent to a coset action, and now we will show that every vertex-transitive graph or digraph can be represented as a so called coset
More informationCompatible Circuit Decompositions of 4-Regular Graphs
Compatible Circuit Decompositions of 4-Regular Graphs Herbert Fleischner, François Genest and Bill Jackson Abstract A transition system T of an Eulerian graph G is a family of partitions of the edges incident
More information8.5 Sequencing Problems. Chapter 8. NP and Computational Intractability. Hamiltonian Cycle. Hamiltonian Cycle
Chapter 8 NP and Computational Intractability 8.5 Sequencing Problems Basic genres. Packing problems: SET-PACKING, INDEPENDENT SET. Covering problems: SET-COVER, VERTEX-COVER. Constraint satisfaction problems:
More informationHamilton Cycles in Digraphs of Unitary Matrices
Hamilton Cycles in Digraphs of Unitary Matrices G. Gutin A. Rafiey S. Severini A. Yeo Abstract A set S V is called an q + -set (q -set, respectively) if S has at least two vertices and, for every u S,
More informationFixed Parameter Algorithms for Interval Vertex Deletion and Interval Completion Problems
Fixed Parameter Algorithms for Interval Vertex Deletion and Interval Completion Problems Arash Rafiey Department of Informatics, University of Bergen, Norway arash.rafiey@ii.uib.no Abstract We consider
More informationInduced subgraphs of graphs with large chromatic number. IX. Rainbow paths
Induced subgraphs of graphs with large chromatic number. IX. Rainbow paths Alex Scott Oxford University, Oxford, UK Paul Seymour 1 Princeton University, Princeton, NJ 08544, USA January 20, 2017; revised
More informationNonnegative Matrices I
Nonnegative Matrices I Daisuke Oyama Topics in Economic Theory September 26, 2017 References J. L. Stuart, Digraphs and Matrices, in Handbook of Linear Algebra, Chapter 29, 2006. R. A. Brualdi and H. J.
More informationAutomorphism groups of wreath product digraphs
Automorphism groups of wreath product digraphs Edward Dobson Department of Mathematics and Statistics Mississippi State University PO Drawer MA Mississippi State, MS 39762 USA dobson@math.msstate.edu Joy
More informationHamiltonian paths in tournaments A generalization of sorting DM19 notes fall 2006
Hamiltonian paths in tournaments A generalization of sorting DM9 notes fall 2006 Jørgen Bang-Jensen Imada, SDU 30. august 2006 Introduction and motivation Tournaments which we will define mathematically
More informationThe Inclusion Exclusion Principle
The Inclusion Exclusion Principle 1 / 29 Outline Basic Instances of The Inclusion Exclusion Principle The General Inclusion Exclusion Principle Counting Derangements Counting Functions Stirling Numbers
More informationSpanning Paths in Infinite Planar Graphs
Spanning Paths in Infinite Planar Graphs Nathaniel Dean AT&T, ROOM 2C-415 600 MOUNTAIN AVENUE MURRAY HILL, NEW JERSEY 07974-0636, USA Robin Thomas* Xingxing Yu SCHOOL OF MATHEMATICS GEORGIA INSTITUTE OF
More informationSimplification by Truth Table and without Truth Table
Engineering Mathematics 2013 SUBJECT NAME SUBJECT CODE MATERIAL NAME MATERIAL CODE REGULATION UPDATED ON : Discrete Mathematics : MA2265 : University Questions : SKMA1006 : R2008 : August 2013 Name of
More information