Perfect 1-factorizations
|
|
- Joshua Long
- 5 years ago
- Views:
Transcription
1 Perfect 1-factorizations Alexander Rosa Department of Mathematics and Statistics, McMaster University Rosa (McMaster) Perfect 1-factorizations 1 / 17
2 Smolenice 1963 Rosa (McMaster) Perfect 1-factorizations 2 / 17
3 Problem No.20 Does there exist an integer n > 1 such that the complete 2n-gon is not Hamiltonian? 3 3 Hamiltonian in the sense of Kotzig, see p.63. Translated into contemporary terminology: Does a perfect 1-factorization of the complete graph K 2n for all integers n > 1? exist Rosa (McMaster) Perfect 1-factorizations 3 / 17
4 Problem No.20 Does there exist an integer n > 1 such that the complete 2n-gon is not Hamiltonian? 3 3 Hamiltonian in the sense of Kotzig, see p.63. Translated into contemporary terminology: Does a perfect 1-factorization of the complete graph K 2n for all integers n > 1? exist Rosa (McMaster) Perfect 1-factorizations 3 / 17
5 Two results of Kotzig on perfect 1-factorizations A 1-factorization of K 2n is perfect if the union of any two of its 1-factors is a Hamiltonian cycle. Theorem 1. A perfect 1-factorization of K 2n exists whenever 2n 1 is a prime. Theorem 2. A perfect 1-factorization of K 2n exists whenever n is a prime. Rosa (McMaster) Perfect 1-factorizations 4 / 17
6 Small orders The values of 2n 100 for which neither 2n 1 nor n is a prime, are n = 16, 28, 36, 40, 50, 52, and 56, 64, 66, 70, 76, 78, 88, 92, 96, 100. After considerable effort, both Kotzig and (independently) Bruce Anderson succeeded in finding a perfect 1-factorization of K 16. Bruce Anderson subsequently constructed a perfect 1-factorization of K 28. A discovery of a perfect 1-factorization for each of the next four orders (n = 36, 40, 50, 52) merited a separate paper! And n = 56 is currently the first open order... Rosa (McMaster) Perfect 1-factorizations 5 / 17
7 Small orders The values of 2n 100 for which neither 2n 1 nor n is a prime, are n = 16, 28, 36, 40, 50, 52, and 56, 64, 66, 70, 76, 78, 88, 92, 96, 100. After considerable effort, both Kotzig and (independently) Bruce Anderson succeeded in finding a perfect 1-factorization of K 16. Bruce Anderson subsequently constructed a perfect 1-factorization of K 28. A discovery of a perfect 1-factorization for each of the next four orders (n = 36, 40, 50, 52) merited a separate paper! And n = 56 is currently the first open order... Rosa (McMaster) Perfect 1-factorizations 5 / 17
8 Small orders The values of 2n 100 for which neither 2n 1 nor n is a prime, are n = 16, 28, 36, 40, 50, 52, and 56, 64, 66, 70, 76, 78, 88, 92, 96, 100. After considerable effort, both Kotzig and (independently) Bruce Anderson succeeded in finding a perfect 1-factorization of K 16. Bruce Anderson subsequently constructed a perfect 1-factorization of K 28. A discovery of a perfect 1-factorization for each of the next four orders (n = 36, 40, 50, 52) merited a separate paper! And n = 56 is currently the first open order... Rosa (McMaster) Perfect 1-factorizations 5 / 17
9 Small orders The values of 2n 100 for which neither 2n 1 nor n is a prime, are n = 16, 28, 36, 40, 50, 52, and 56, 64, 66, 70, 76, 78, 88, 92, 96, 100. After considerable effort, both Kotzig and (independently) Bruce Anderson succeeded in finding a perfect 1-factorization of K 16. Bruce Anderson subsequently constructed a perfect 1-factorization of K 28. A discovery of a perfect 1-factorization for each of the next four orders (n = 36, 40, 50, 52) merited a separate paper! And n = 56 is currently the first open order... Rosa (McMaster) Perfect 1-factorizations 5 / 17
10 Small orders The values of 2n 100 for which neither 2n 1 nor n is a prime, are n = 16, 28, 36, 40, 50, 52, and 56, 64, 66, 70, 76, 78, 88, 92, 96, 100. After considerable effort, both Kotzig and (independently) Bruce Anderson succeeded in finding a perfect 1-factorization of K 16. Bruce Anderson subsequently constructed a perfect 1-factorization of K 28. A discovery of a perfect 1-factorization for each of the next four orders (n = 36, 40, 50, 52) merited a separate paper! And n = 56 is currently the first open order... Rosa (McMaster) Perfect 1-factorizations 5 / 17
11 Difficulties Why is it so difficult to find perfect 1-factorizations? Part of the answer: n of (n) A pof (n) A of (n) = number of nonisomorphic 1-factorizations of K 2n, pof (n) = number of nonisomorphic perfect 1-factorizations of K 2n Rosa (McMaster) Perfect 1-factorizations 6 / 17
12 Difficulties Why is it so difficult to find perfect 1-factorizations? Part of the answer: n of (n) A pof (n) A of (n) = number of nonisomorphic 1-factorizations of K 2n, pof (n) = number of nonisomorphic perfect 1-factorizations of K 2n Rosa (McMaster) Perfect 1-factorizations 6 / 17
13 Ihrig s results In a series of papers, Ed Ihrig proves several results on the structure of automorphism groups of perfect 1-factorizations. E.g.: If F is a starter-induced perfect 1-factorization of K 2n then the automorphism group of F is a semidirect product of Z 2n 1 with H where H is a subgroup of the automorphism group of Z 2n 1, H divides n 1 and H is odd. For example, Ihrig, Seah and Stinson find a perfect 1-factorization of K 50 by searching for a starter in Z 49 which is fixed by a multiplicative subgroup {1, 18, 30} (i.e., assuming as an automorphism group the semidirect product of Z 49 with Z 3 ). Rosa (McMaster) Perfect 1-factorizations 7 / 17
14 Ihrig s results In a series of papers, Ed Ihrig proves several results on the structure of automorphism groups of perfect 1-factorizations. E.g.: If F is a starter-induced perfect 1-factorization of K 2n then the automorphism group of F is a semidirect product of Z 2n 1 with H where H is a subgroup of the automorphism group of Z 2n 1, H divides n 1 and H is odd. For example, Ihrig, Seah and Stinson find a perfect 1-factorization of K 50 by searching for a starter in Z 49 which is fixed by a multiplicative subgroup {1, 18, 30} (i.e., assuming as an automorphism group the semidirect product of Z 49 with Z 3 ). Rosa (McMaster) Perfect 1-factorizations 7 / 17
15 Recursive constructions? Another drawback: all known "sporadic" P1Fs were obtained by direct constructions, mostly using properties of finite fields. No recursive methods are known. Attempts to obtain a 2n 4n 2 construction using P1Fs generated by starters in cyclic groups have not been successful so far. Rosa (McMaster) Perfect 1-factorizations 8 / 17
16 Conclusion Kotzig s conjecture A PERFECT 1-FACTORIZATION OF K 2n EXISTS FOR ALL n > 1 is likely to remain open for quite some time... Rosa (McMaster) Perfect 1-factorizations 9 / 17
17 Perfect 1-factorizations of cubic graphs We consider only cubic graphs of Class 1. These fall into three categories. Category N. Category P. Cubic graphs without a P1F. Cubic graphs admitting only P1Fs. Category PN. Cubic graphs admitting a P1F and also a non-p1f. Rosa (McMaster) Perfect 1-factorizations 10 / 17
18 Forbidden subgraphs Some subgraphs cannot occur in cubic graphs admitting P1F: K 4 e, G 1, G 2 G 1 G 2 Rosa (McMaster) Perfect 1-factorizations 11 / 17
19 Category P: Cubic graphs admitting only P1Fs. Rosa (McMaster) Perfect 1-factorizations 12 / 17
20 Category PN. Cubic graphs admitting a P1F and also a non-p1f. We may assume that the number of vertices is at least 8. There are some forbidden subgraphs of cubic graphs admitting a P1F. Rosa (McMaster) Perfect 1-factorizations 13 / 17
21 Kotzig s theorem. A bipartite cubic graph with 2n vertices admits a perfect 1-factorization only if n is odd. For example, the prism GP(2k, 1) (prism with 4k vertices) cannot have a P1F (but obviously has a 1-factorization) Cubic graphs of category P exist for all orders 2n 4. (Examples of cubic graphs with a unique 1-factorization.). Number of vertices Class N P PN Total Rosa (McMaster) Perfect 1-factorizations 14 / 17
22 Generalized Petersen graphs GP(n, k) These are cubic graphs with vertex set Z n {0, 1} and edge-set {i 0, (i + 1) 0 }, {i 1, (i + k) 1 }, {i 0, i 1 }, i Z n (n 5, k < n 2 ). Every generalized Petersen graph, except GP(5, 2) (the Petersen graph itself), is Class 1. Bonvicini-Mazzuoccolo: GP(n, 2) has a P1F if and only if n 3 or 4 (mod 6). GP(n, 3) has a P1F if and only if n = 9. GP(n, k) does not admit P1F when n is even and k is odd. For k 4, to determine whether GP(n, k) admits a P1F remains open. Rosa (McMaster) Perfect 1-factorizations 15 / 17
23 GP(n, 4) and GP(n, 5) The graph GP(9, 4) is the well-known Tutte s example of a uniquely 3-edge-colourable cubic graph, and so is category P. When n is odd and 9 n 65, then GP(n, 4) has a P1F, except when n {11, 13, 17, 35}. When n is even and 10 n 64, then GP(n, 4) has a P1F if and only if n {28, 30} or 46 n 64. GP(n, 5) does not have a P1F when n is even. But when n is odd, and 11 n 51 then GP(n, 5) admits a P1F except when n {11, 13, 29, 41}. Moreover, GP(15, 5) admits only perfect 1-factorizations! It is difficult to even formulate an existence conjecture for GP(n, 4) or GP(n, 5). Rosa (McMaster) Perfect 1-factorizations 16 / 17
24 GP(n, 4) and GP(n, 5) The graph GP(9, 4) is the well-known Tutte s example of a uniquely 3-edge-colourable cubic graph, and so is category P. When n is odd and 9 n 65, then GP(n, 4) has a P1F, except when n {11, 13, 17, 35}. When n is even and 10 n 64, then GP(n, 4) has a P1F if and only if n {28, 30} or 46 n 64. GP(n, 5) does not have a P1F when n is even. But when n is odd, and 11 n 51 then GP(n, 5) admits a P1F except when n {11, 13, 29, 41}. Moreover, GP(15, 5) admits only perfect 1-factorizations! It is difficult to even formulate an existence conjecture for GP(n, 4) or GP(n, 5). Rosa (McMaster) Perfect 1-factorizations 16 / 17
25 GP(n, 4) and GP(n, 5) The graph GP(9, 4) is the well-known Tutte s example of a uniquely 3-edge-colourable cubic graph, and so is category P. When n is odd and 9 n 65, then GP(n, 4) has a P1F, except when n {11, 13, 17, 35}. When n is even and 10 n 64, then GP(n, 4) has a P1F if and only if n {28, 30} or 46 n 64. GP(n, 5) does not have a P1F when n is even. But when n is odd, and 11 n 51 then GP(n, 5) admits a P1F except when n {11, 13, 29, 41}. Moreover, GP(15, 5) admits only perfect 1-factorizations! It is difficult to even formulate an existence conjecture for GP(n, 4) or GP(n, 5). Rosa (McMaster) Perfect 1-factorizations 16 / 17
26 GP(n, 4) and GP(n, 5) The graph GP(9, 4) is the well-known Tutte s example of a uniquely 3-edge-colourable cubic graph, and so is category P. When n is odd and 9 n 65, then GP(n, 4) has a P1F, except when n {11, 13, 17, 35}. When n is even and 10 n 64, then GP(n, 4) has a P1F if and only if n {28, 30} or 46 n 64. GP(n, 5) does not have a P1F when n is even. But when n is odd, and 11 n 51 then GP(n, 5) admits a P1F except when n {11, 13, 29, 41}. Moreover, GP(15, 5) admits only perfect 1-factorizations! It is difficult to even formulate an existence conjecture for GP(n, 4) or GP(n, 5). Rosa (McMaster) Perfect 1-factorizations 16 / 17
27 GP(n, 4) and GP(n, 5) The graph GP(9, 4) is the well-known Tutte s example of a uniquely 3-edge-colourable cubic graph, and so is category P. When n is odd and 9 n 65, then GP(n, 4) has a P1F, except when n {11, 13, 17, 35}. When n is even and 10 n 64, then GP(n, 4) has a P1F if and only if n {28, 30} or 46 n 64. GP(n, 5) does not have a P1F when n is even. But when n is odd, and 11 n 51 then GP(n, 5) admits a P1F except when n {11, 13, 29, 41}. Moreover, GP(15, 5) admits only perfect 1-factorizations! It is difficult to even formulate an existence conjecture for GP(n, 4) or GP(n, 5). Rosa (McMaster) Perfect 1-factorizations 16 / 17
28 THANK YOU FOR YOUR ATTENTION! Rosa (McMaster) Perfect 1-factorizations 17 / 17
Automorphism groups of Steiner triple systems
Automorphism groups of Steiner triple systems Alexander Rosa Department of Mathematics and Statistics, McMaster University Rosa (McMaster) Automorphism groups of Steiner triple systems 1 / 20 Steiner triple
More informationColourings of cubic graphs inducing isomorphic monochromatic subgraphs
Colourings of cubic graphs inducing isomorphic monochromatic subgraphs arxiv:1705.06928v2 [math.co] 10 Sep 2018 Marién Abreu 1, Jan Goedgebeur 2, Domenico Labbate 1, Giuseppe Mazzuoccolo 3 1 Dipartimento
More informationarxiv: v1 [math.co] 4 Jan 2018
A family of multigraphs with large palette index arxiv:80.0336v [math.co] 4 Jan 208 M.Avesani, A.Bonisoli, G.Mazzuoccolo July 22, 208 Abstract Given a proper edge-coloring of a loopless multigraph, the
More informationOn Hamiltonian cycle systems with a nice automorphism group
On Hamiltonian cycle systems with a nice automorphism group Francesca Merola Università Roma Tre April Hamiltonian cycle systems set B = {C,..., C s } of n-cycles of Γ such that the edges E(C ),... E(C
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 informationAlgebraically defined graphs and generalized quadrangles
Department of Mathematics Kutztown University of Pennsylvania Combinatorics and Computer Algebra 2015 July 22, 2015 Cages and the Moore bound For given positive integers k and g, find the minimum number
More informationNowhere zero flow. Definition: A flow on a graph G = (V, E) is a pair (D, f) such that. 1. D is an orientation of G. 2. f is a function on E.
Nowhere zero flow Definition: A flow on a graph G = (V, E) is a pair (D, f) such that 1. D is an orientation of G. 2. f is a function on E. 3. u N + D (v) f(uv) = w ND f(vw) for every (v) v V. Example:
More informationCycle decompositions of the complete graph
Cycle decompositions of the complete graph A.J.W. Hilton Department of Mathematics University of Reading Whiteknights P.O. Box 220 Reading RG6 6AX U.K. Matthew Johnson Department of Mathematics London
More informationMonogamous decompositions of complete bipartite graphs, symmetric H -squares, and self-orthogonal I-factorizations
Monogamous decompositions of complete bipartite graphs, symmetric H -squares, and self-orthogonal I-factorizations C.C. Lindner Department of Discrete and Statistical Sciences Auburn University Auburn,
More informationInduced Cycles of Fixed Length
Induced Cycles of Fixed Length Terry McKee Wright State University Dayton, Ohio USA terry.mckee@wright.edu Cycles in Graphs Vanderbilt University 31 May 2012 Overview 1. Investigating the fine structure
More informationColourings of cubic graphs inducing isomorphic monochromatic subgraphs
Colourings of cubic graphs inducing isomorphic monochromatic subgraphs arxiv:1705.06928v1 [math.co] 19 May 2017 Marién Abreu 1, Jan Goedgebeur 2, Domenico Labbate 1, Giuseppe Mazzuoccolo 3 1 Dipartimento
More informationCubic Cayley Graphs and Snarks
Cubic Cayley Graphs and Snarks Klavdija Kutnar University of Primorska Nashville, 2012 Snarks A snark is a connected, bridgeless cubic graph with chromatic index equal to 4. non-snark = bridgeless cubic
More informationHAMILTONICITY OF VERTEX-TRANSITIVE GRAPHS
HAMILTONICITY OF VERTEX-TRANSITIVE GRAPHS University of Primorska & University of Ljubljana June, 2011 Announcing two opening positions at UP Famnit / Pint in 2011/12 Young Research position = PhD Grant
More informationSome New Perfect One-Factorizations from Starters in Finite Fields
Some New Perfect One-Factorizations from Starters in Finite Fields J.H. Dinitz UNlVERSlTY OF VERMONT D. R. Stinson UNlVERSlTY OF MANlTOBA ABSTRACT We construct seven new examples of perfect one-factorizations,
More informationA DISCOURSE ON THREE COMBINATORIAL DISEASES. Alexander Rosa
A DISCOURSE ON THREE COMBINATORIAL DISEASES Alexander Rosa Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada Frank Harary in his article [2] was the first to speak
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 informationPerfect matchings in highly cyclically connected regular graphs
Perfect matchings in highly cyclically connected regular graphs arxiv:1709.08891v1 [math.co] 6 Sep 017 Robert Lukot ka Comenius University, Bratislava lukotka@dcs.fmph.uniba.sk Edita Rollová University
More informationM-saturated M={ } M-unsaturated. Perfect Matching. Matchings
Matchings A matching M of a graph G = (V, E) is a set of edges, no two of which are incident to a common vertex. M-saturated M={ } M-unsaturated Perfect Matching 1 M-alternating path M not M M not M M
More informationSelf-complementary circulant graphs
Self-complementary circulant graphs Brian Alspach Joy Morris Department of Mathematics and Statistics Burnaby, British Columbia Canada V5A 1S6 V. Vilfred Department of Mathematics St. Jude s College Thoothoor
More informationSome Applications of pq-groups in Graph Theory
Some Applications of pq-groups in Graph Theory Geoffrey Exoo Department of Mathematics and Computer Science Indiana State University Terre Haute, IN 47809 g-exoo@indstate.edu January 25, 2002 Abstract
More informationSOME QUESTIONS ARISING FROM THE STUDY OF THE GENERATING GRAPH
SOME QUESTIONS ARISING FROM THE STUDY OF Andrea Lucchini Università di Padova, Italy ISCHIA GROUP THEORY 2012 March, 26th - 29th The generating graph Γ(G) of a group G is the graph defined on the non-identity
More informationPrimitive 2-factorizations of the complete graph
Discrete Mathematics 308 (2008) 175 179 www.elsevier.com/locate/disc Primitive 2-factorizations of the complete graph Giuseppe Mazzuoccolo,1 Dipartimento di Matematica, Università di Modena e Reggio Emilia,
More informationGraph products and new solutions to Oberwolfach problems
Graph products and new solutions to Oberwolfach problems Gloria Rinaldi Dipartimento di Scienze e Metodi dell Ingegneria Università di Modena e Reggio Emilia 42100 Reggio Emilia, Italy gloria.rinaldi@unimore.it
More informationDIRECTED CYCLIC HAMILTONIAN CYCLE SYSTEMS OF THE COMPLETE SYMMETRIC DIGRAPH
DIRECTED CYCLIC HAMILTONIAN CYCLE SYSTEMS OF THE COMPLETE SYMMETRIC DIGRAPH HEATHER JORDON AND JOY MORRIS Abstract. In this paper, we prove that directed cyclic hamiltonian cycle systems of the complete
More informationCycles through 23 vertices in 3-connected cubic planar graphs
Cycles through 23 vertices in 3-connected cubic planar graphs R. E. L. Aldred, S. Bau and D. A. Holton Department of Mathematics and Statistics, University of Otago, P.O. Box 56, Dunedin, New Zealand.
More informationGraphs & Algorithms: Advanced Topics Nowhere-Zero Flows
Graphs & Algorithms: Advanced Topics Nowhere-Zero Flows Uli Wagner ETH Zürich Flows Definition Let G = (V, E) be a multigraph (allow loops and parallel edges). An (integer-valued) flow on G (also called
More informationOn monomial graphs of girth 8.
On monomial graphs of girth 8. Vasyl Dmytrenko, Felix Lazebnik, Jason Williford July 17, 2009 Let q = p e, f F q [x] is a permutation polynomial on F q (PP) if c f(c) is a bijection of F q to F q. Examples:
More informationOn Hamilton Decompositions of Infinite Circulant Graphs
On Hamilton Decompositions of Infinite Circulant Graphs Darryn Bryant 1, Sarada Herke 1, Barbara Maenhaut 1, and Bridget Webb 2 1 School of Mathematics and Physics, The University of Queensland, QLD 4072,
More informationOn the Existence of a Second Hamilton Cycle in Hamiltonian Graphs With Symmetry
On the Existence of a Second Hamilton Cycle in Hamiltonian Graphs With Symmetry Andrew Wagner Thesis submitted to the Faculty of Graduate and Postdoctoral Studies in partial fulfillment of the requirements
More informationSemiregular automorphisms of vertex-transitive graphs
Semiregular automorphisms of vertex-transitive graphs Michael Giudici http://www.maths.uwa.edu.au/ giudici/research.html Semiregular automorphisms A semiregular automorphism of a graph is a nontrivial
More informationA Family of Perfect Factorisations of Complete Bipartite Graphs
Journal of Combinatorial Theory, Series A 98, 328 342 (2002) doi:10.1006/jcta.2001.3240, available online at http://www.idealibrary.com on A Family of Perfect Factorisations of Complete Bipartite Graphs
More informationCycle Double Covers and Semi-Kotzig Frame
Cycle Double Covers and Semi-Kotzig Frame Dong Ye and Cun-Quan Zhang arxiv:1105.5190v1 [math.co] 26 May 2011 Department of Mathematics, West Virginia University, Morgantown, WV 26506-6310 Emails: dye@math.wvu.edu;
More information1-factor and cycle covers of cubic graphs
1-factor and cycle covers of cubic graphs arxiv:1209.4510v [math.co] 29 Jan 2015 Eckhard Steffen Abstract Let G be a bridgeless cubic graph. Consider a list of k 1-factors of G. Let E i be the set of edges
More information0-Sum and 1-Sum Flows in Regular Graphs
0-Sum and 1-Sum Flows in Regular Graphs S. Akbari Department of Mathematical Sciences Sharif University of Technology Tehran, Iran School of Mathematics, Institute for Research in Fundamental Sciences
More informationUnions of perfect matchings in cubic graphs
Unions of perfect matchings in cubic graphs Tomáš Kaiser Daniel Král Serguei Norine Abstract We show that any cubic bridgeless graph with m edges contains two perfect matchings that cover at least 3m/5
More informationSymmetric bowtie decompositions of the complete graph
Symmetric bowtie decompositions of the complete graph Simona Bonvicini Dipartimento di Scienze e Metodi dell Ingegneria Via Amendola, Pad. Morselli, 4100 Reggio Emilia, Italy simona.bonvicini@unimore.it
More informationMultiple Petersen subdivisions in permutation graphs
Multiple Petersen subdivisions in permutation graphs arxiv:1204.1989v1 [math.co] 9 Apr 2012 Tomáš Kaiser 1 Jean-Sébastien Sereni 2 Zelealem Yilma 3 Abstract A permutation graph is a cubic graph admitting
More informationColoring. Basics. A k-coloring of a loopless graph G is a function f : V (G) S where S = k (often S = [k]).
Coloring Basics A k-coloring of a loopless graph G is a function f : V (G) S where S = k (often S = [k]). For an i S, the set f 1 (i) is called a color class. A k-coloring is called proper if adjacent
More informationOn Perfect Matching Coverings and Even Subgraph Coverings
On Perfect Matching Coverings and Even Subgraph Coverings Xinmin Hou, Hong-Jian Lai, 2 and Cun-Quan Zhang 2 SCHOOL OF MATHEMATICAL SCIENCES UNIVERSITY OF SCIENCE AND TECHNOLOGY OF CHINA HEFEI, ANHUI, 20026
More informationarxiv: v1 [math.co] 21 Jan 2016
Cores, joins and the Fano-flow conjectures Ligang Jin, Giuseppe Mazzuoccolo, Eckhard Steffen August 13, 2018 arxiv:1601.05762v1 [math.co] 21 Jan 2016 Abstract The Fan-Raspaud Conjecture states that every
More informationOn the Dynamic Chromatic Number of Graphs
On the Dynamic Chromatic Number of Graphs Maryam Ghanbari Joint Work with S. Akbari and S. Jahanbekam Sharif University of Technology m_phonix@math.sharif.ir 1. Introduction Let G be a graph. A vertex
More informationInduction. Induction. Induction. Induction. Induction. Induction 2/22/2018
The principle of mathematical induction is a useful tool for proving that a certain predicate is true for all natural numbers. It cannot be used to discover theorems, but only to prove them. If we have
More informationCubic Cayley graphs and snarks
Cubic Cayley graphs and snarks University of Primorska UP FAMNIT, Feb 2012 Outline I. Snarks II. Independent sets in cubic graphs III. Non-existence of (2, s, 3)-Cayley snarks IV. Snarks and (2, s, t)-cayley
More informationOn Barnette's Conjecture. Jens M. Schmidt
On Barnette's Conjecture Jens M. Schmidt Hamiltonian Cycles Def. A graph is Hamiltonian if it contains a Hamiltonian cycle, i.e., a cycle that contains every vertex exactly once. William R. Hamilton 3-Connectivity
More informationIndependent Dominating Sets and a Second Hamiltonian Cycle in Regular Graphs
Journal of Combinatorial Theory, Series B 72, 104109 (1998) Article No. TB971794 Independent Dominating Sets and a Second Hamiltonian Cycle in Regular Graphs Carsten Thomassen Department of Mathematics,
More informationZero-Sum Flows in Regular Graphs
Zero-Sum Flows in Regular Graphs S. Akbari,5, A. Daemi 2, O. Hatami, A. Javanmard 3, A. Mehrabian 4 Department of Mathematical Sciences Sharif University of Technology Tehran, Iran 2 Department of Mathematics
More informationThe path of order 5 is called the W-graph. Let (X, Y ) be a bipartition of the W-graph with X = 3. Vertices in X are called big vertices, and vertices
The Modular Chromatic Number of Trees Jia-Lin Wang, Jie-Ying Yuan Abstract:The concept of modular coloring and modular chromatic number was proposed by F. Okamoto, E. Salehi and P. Zhang in 2010. They
More informationA Characterization of Graphs with Fractional Total Chromatic Number Equal to + 2
A Characterization of Graphs with Fractional Total Chromatic Number Equal to + Takehiro Ito a, William. S. Kennedy b, Bruce A. Reed c a Graduate School of Information Sciences, Tohoku University, Aoba-yama
More informationRegular embeddings of graphs into surfaces
Banská Bystrica Wien, November 28 November 2, 28 Maps and graph embeddings Maps Map is a 2-cell decomposition of a surface Category OrMaps Maps on orientable surfaces together with orientation-preserving
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 informationHamiltonicity in Connected Regular Graphs
Hamiltonicity in Connected Regular Graphs Daniel W. Cranston Suil O April 29, 2012 Abstract In 1980, Jackson proved that every 2-connected k-regular graph with at most 3k vertices is Hamiltonian. This
More informationDirected cyclic Hamiltonian cycle systems of the complete symmetric digraph
Discrete Mathematics 309 (2009) 784 796 www.elsevier.com/locate/disc Directed cyclic Hamiltonian cycle systems of the complete symmetric digraph Heather Jordon a,, Joy Morris b a Department of Mathematics,
More information11 Block Designs. Linear Spaces. Designs. By convention, we shall
11 Block Designs Linear Spaces In this section we consider incidence structures I = (V, B, ). always let v = V and b = B. By convention, we shall Linear Space: We say that an incidence structure (V, B,
More informationA note on the Isomorphism Problem for Monomial Digraphs
A note on the Isomorphism Problem for Monomial Digraphs Aleksandr Kodess Department of Mathematics University of Rhode Island kodess@uri.edu Felix Lazebnik Department of Mathematical Sciences University
More informationKevin James. MTHSC 412 Section 3.4 Cyclic Groups
MTHSC 412 Section 3.4 Cyclic Groups Definition If G is a cyclic group and G =< a > then a is a generator of G. Definition If G is a cyclic group and G =< a > then a is a generator of G. Example 1 Z is
More informationZero sum partition of Abelian groups into sets of the same order and its applications
Zero sum partition of Abelian groups into sets of the same order and its applications Sylwia Cichacz Faculty of Applied Mathematics AGH University of Science and Technology Al. Mickiewicza 30, 30-059 Kraków,
More informationGroup connectivity of certain graphs
Group connectivity of certain graphs Jingjing Chen, Elaine Eschen, Hong-Jian Lai May 16, 2005 Abstract Let G be an undirected graph, A be an (additive) Abelian group and A = A {0}. A graph G is A-connected
More informationThe Complexity of Computing the Sign of the Tutte Polynomial
The Complexity of Computing the Sign of the Tutte Polynomial Leslie Ann Goldberg (based on joint work with Mark Jerrum) Oxford Algorithms Workshop, October 2012 The Tutte polynomial of a graph G = (V,
More informationSkolem-type Difference Sets for Cycle Systems
Skolem-type Difference Sets for Cycle Systems Darryn Bryant Department of Mathematics University of Queensland Qld 072 Australia Heather Gavlas Department of Mathematics Illinois State University Campus
More informationSymmetries That Latin Squares Inherit from 1-Factorizations
Symmetries That Latin Squares Inherit from 1-Factorizations Ian M. Wanless, 1 Edwin C. Ihrig 2 1 Department of Computer Science, Australian National University, ACT 0200 Australia, E-mail: imw@cs.anu.edu.au
More informationOut-colourings of Digraphs
Out-colourings of Digraphs N. Alon J. Bang-Jensen S. Bessy July 13, 2017 Abstract We study vertex colourings of digraphs so that no out-neighbourhood is monochromatic and call such a colouring an out-colouring.
More informationHamiltonian decomposition of prisms over cubic graphs
Hamiltonian decomposition of prisms over cubic graphs Moshe Rosenfeld, Ziqing Xiang To cite this version: Moshe Rosenfeld, Ziqing Xiang. Hamiltonian decomposition of prisms over cubic graphs. Discrete
More informationHamiltonian embeddings from triangulations
Hamiltonian embeddings from triangulations Mike J Grannell, Terry S Griggs Department of Mathematics The Open University Walton Hall Milton Keynes MK7 6AA UK Jozef Širáň Department of Mathematics Faculty
More informationMATH CSE20 Homework 5 Due Monday November 4
MATH CSE20 Homework 5 Due Monday November 4 Assigned reading: NT Section 1 (1) Prove the statement if true, otherwise find a counterexample. (a) For all natural numbers x and y, x + y is odd if one of
More informationEven Pairs and Prism Corners in Square-Free Berge Graphs
Even Pairs and Prism Corners in Square-Free Berge Graphs Maria Chudnovsky Princeton University, Princeton, NJ 08544 Frédéric Maffray CNRS, Laboratoire G-SCOP, University of Grenoble-Alpes, France Paul
More informationCORES, JOINS AND THE FANO-FLOW CONJECTURES
Discussiones Mathematicae Graph Theory 38 (2018) 165 175 doi:10.7151/dmgt.1999 CORES, JOINS AND THE FANO-FLOW CONJECTURES Ligang Jin, Eckhard Steffen Paderborn University Institute of Mathematics and Paderborn
More informationExtendability of Contractible Configurations for Nowhere-Zero Flows and Modulo Orientations
Graphs and Combinatorics (2016) 32:1065 1075 DOI 10.1007/s00373-015-1636-0 ORIGINAL PAPER Extendability of Contractible Configurations for Nowhere-Zero Flows and Modulo Orientations Yanting Liang 1 Hong-Jian
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 informationConnectivity of Cages
Connectivity of Cages H. L. Fu, 1,2 1 DEPARTMENT OF APPLIED MATHEMATICS NATIONAL CHIAO-TUNG UNIVERSITY HSIN-CHU, TAIWAN REPUBLIC OF CHINA K. C. Huang, 3 3 DEPARTMENT OF APPLIED MATHEMATICS PROVIDENCE UNIVERSITY,
More informationAdvanced Combinatorial Optimization September 22, Lecture 4
8.48 Advanced Combinatorial Optimization September 22, 2009 Lecturer: Michel X. Goemans Lecture 4 Scribe: Yufei Zhao In this lecture, we discuss some results on edge coloring and also introduce the notion
More informationSemiregular automorphisms of vertex-transitive cubic graphs
Semiregular automorphisms of vertex-transitive cubic graphs Peter Cameron a,1 John Sheehan b Pablo Spiga a a School of Mathematical Sciences, Queen Mary, University of London, Mile End Road, London E1
More informationCYCLE STRUCTURES IN GRAPHS. Angela K. Harris. A thesis submitted to the. University of Colorado Denver. in partial fulfillment
CYCLE STRUCTURES IN GRAPHS by Angela K. Harris Master of Science, University of South Alabama, 003 A thesis submitted to the University of Colorado Denver in partial fulfillment of the requirements for
More informationMINIMALLY NON-PFAFFIAN GRAPHS
MINIMALLY NON-PFAFFIAN GRAPHS SERGUEI NORINE AND ROBIN THOMAS Abstract. We consider the question of characterizing Pfaffian graphs. We exhibit an infinite family of non-pfaffian graphs minimal with respect
More informationUNIVERSITÀ DEGLI STUDI DI ROMA LA SAPIENZA
UNIVERSITÀ DEGLI STUDI DI ROMA LA SAPIENZA On d-graceful labelings Anita Pasotti Quaderni Elettronici del Seminario di Geometria Combinatoria 7E (Maggio 0) http://wwwmatuniromait/~combinat/quaderni Dipartimento
More informationUnions of perfect matchings in cubic graphs
Unions of perfect matchings in cubic graphs Tomáš Kaiser 1,2 Department of Mathematics and Institute for Theoretical Computer Science (ITI) University of West Bohemia Univerzitní 8, 306 14 Plzeň, Czech
More information1 Introductory remarks Throughout this paper graphs are nite, simple and undirected. Adopting the terminology of Tutte [11], a k-arc in a graph X is a
ON 2-ARC-TRANSITIVE CAYLEY GRAPHS OF DIHEDRAL GROUPS Dragan Marusic 1 IMFM, Oddelek za matematiko Univerza v Ljubljani Jadranska 19, 1111 Ljubljana Slovenija Abstract A partial extension of the results
More informationGeneralized Pigeonhole Properties of Graphs and Oriented Graphs
Europ. J. Combinatorics (2002) 23, 257 274 doi:10.1006/eujc.2002.0574 Available online at http://www.idealibrary.com on Generalized Pigeonhole Properties of Graphs and Oriented Graphs ANTHONY BONATO, PETER
More informationOn (k, d)-multiplicatively indexable graphs
Chapter 3 On (k, d)-multiplicatively indexable graphs A (p, q)-graph G is said to be a (k,d)-multiplicatively indexable graph if there exist an injection f : V (G) N such that the induced function f :
More informationTough graphs and hamiltonian circuits
Discrete Mathematics 306 (2006) 910 917 www.elsevier.com/locate/disc Tough graphs and hamiltonian circuits V. Chvátal Centre de Recherches Mathématiques, Université de Montréal, Montréal, Canada Abstract
More information9 - The Combinatorial Nullstellensatz
9 - The Combinatorial Nullstellensatz Jacques Verstraëte jacques@ucsd.edu Hilbert s nullstellensatz says that if F is an algebraically closed field and f and g 1, g 2,..., g m are polynomials in F[x 1,
More informationOn Two Unsolved Problems Concerning Matching Covered Graphs
arxiv:1705.09428v1 [math.co] 26 May 2017 On Two Unsolved Problems Concerning Matching Covered Graphs Dedicated to the memory of Professor W.T.Tutte on the occasion of the centennial of his birth Cláudio
More informationDiskrete Mathematik und Optimierung
Diskrete Mathematik und Optimierung Winfried Hochstättler: Towards a flow theory for the dichromatic number Technical Report feu-dmo032.14 Contact: winfried.hochstaettler@fernuni-hagen.de FernUniversität
More informationList of Theorems. Mat 416, Introduction to Graph Theory. Theorem 1 The numbers R(p, q) exist and for p, q 2,
List of Theorems Mat 416, Introduction to Graph Theory 1. Ramsey s Theorem for graphs 8.3.11. Theorem 1 The numbers R(p, q) exist and for p, q 2, R(p, q) R(p 1, q) + R(p, q 1). If both summands on the
More informationLabelings of unions of up to four uniform cycles
AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 29 (2004), Pages 323 336 Labelings of unions of up to four uniform cycles Diane Donovan Centre for Discrete Mathematics and Computing Department of Mathematics
More informationOn Super Edge-magic Total Labeling of Modified Watermill Graph
Journal of Physics: Conference Series PAPER OPEN ACCESS On Super Edge-magic Total Labeling of Modified Watermill Graph To cite this article: Nurdin et al 018 J. Phys.: Conf. Ser. 979 01067 View the article
More informationNotes on Graph Theory
Notes on Graph Theory Maris Ozols June 8, 2010 Contents 0.1 Berge s Lemma............................................ 2 0.2 König s Theorem........................................... 3 0.3 Hall s Theorem............................................
More informationNORTHERN INDIA ENGINEERING COLLEGE, LKO D E P A R T M E N T O F M A T H E M A T I C S. B.TECH IIIrd SEMESTER QUESTION BANK ACADEMIC SESSION
NORTHERN INDIA ENGINEERING COLLEGE, LKO D E P A R T M E N T O F M A T H E M A T I C S B.TECH IIIrd SEMESTER QUESTION BANK ACADEMIC SESSION 011-1 DISCRETE MATHEMATICS (EOE 038) 1. UNIT I (SET, RELATION,
More informationGROUPS AS GRAPHS. W. B. Vasantha Kandasamy Florentin Smarandache
GROUPS AS GRAPHS W. B. Vasantha Kandasamy Florentin Smarandache 009 GROUPS AS GRAPHS W. B. Vasantha Kandasamy e-mail: vasanthakandasamy@gmail.com web: http://mat.iitm.ac.in/~wbv www.vasantha.in Florentin
More informationHamilton weight and Petersen minor
Hamilton weight and Petersen minor Hong-Jian Lai and Cun-Quan Zhang Department of Mathematics West Virginia University Morgantown, WV 26506-6310, USA Email: hjlai@math.wvu.edu, cqzhang@math.wvu.edu Abstract
More informationPrime Factorization and GCF. In my own words
Warm- up Problem What is a prime number? A PRIME number is an INTEGER greater than 1 with EXACTLY 2 positive factors, 1 and the number ITSELF. Examples of prime numbers: 2, 3, 5, 7 What is a composite
More informationSolutions to Assignment 4
1. Let G be a finite, abelian group written additively. Let x = g G g, and let G 2 be the subgroup of G defined by G 2 = {g G 2g = 0}. (a) Show that x = g G 2 g. (b) Show that x = 0 if G 2 = 2. If G 2
More informationChapter 7 Matchings and r-factors
Chapter 7 Matchings and r-factors Section 7.0 Introduction Suppose you have your own company and you have several job openings to fill. Further, suppose you have several candidates to fill these jobs and
More informationMATH 251, Handout on Sylow, Direct Products, and Finite Abelian Groups
MATH 251, Handout on Sylow, Direct Products, and Finite Abelian Groups 1. Let G be a group with 56 elements. Show that G always has a normal subgroup. Solution: Our candidates for normal subgroup are the
More informationGraph coloring, perfect graphs
Lecture 5 (05.04.2013) Graph coloring, perfect graphs Scribe: Tomasz Kociumaka Lecturer: Marcin Pilipczuk 1 Introduction to graph coloring Definition 1. Let G be a simple undirected graph and k a positive
More informationProbe interval graphs and probe unit interval graphs on superclasses of cographs
Author manuscript, published in "" Discrete Mathematics and Theoretical Computer Science DMTCS vol. 15:2, 2013, 177 194 Probe interval graphs and probe unit interval graphs on superclasses of cographs
More informationand critical partial Latin squares.
Nowhere-zero 4-flows, simultaneous edge-colorings, and critical partial Latin squares Rong Luo Department of Mathematical Sciences Middle Tennessee State University Murfreesboro, TN 37132, U.S.A luor@math.wvu.edu
More informationProof of a Conjecture on Monomial Graphs
Proof of a Conjecture on Monomial Graphs Xiang-dong Hou Department of Mathematics and Statistics University of South Florida Joint work with Stephen D. Lappano and Felix Lazebnik New Directions in Combinatorics
More informationDecomposition of Complete Tripartite Graphs Into 5-Cycles
Decomposition of Complete Tripartite Graphs Into 5-Cycles E.S. MAHMOODIAN MARYAM MIRZAKHANI Department of Mathematical Sciences Sharif University of Technology P.O. Box 11365 9415 Tehran, I.R. Iran emahmood@irearn.bitnet
More informationArc-transitive Bicirculants
1 / 27 Arc-transitive Bicirculants Klavdija Kutnar University of Primorska, Slovenia This is a joint work with Iva Antončič, Aubin Arroyo, Isabel Hubard, Ademir Hujdurović, Eugenia O Reilly and Primož
More informationA MATROID EXTENSION RESULT
A MATROID EXTENSION RESULT JAMES OXLEY Abstract. Adding elements to matroids can be fraught with difficulty. In the Vámos matroid V 8, there are four independent sets X 1, X 2, X 3, and X 4 such that (X
More information