On Barnette's Conjecture. Jens M. Schmidt
|
|
|
- Maurice Hart
- 5 years ago
- Views:
Transcription
1 On Barnette's Conjecture Jens M. Schmidt
2 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 3-Connectivity Let G=(V,E) be a simple finite graph, n= V, m= E. Def. G is 3-connected n > 3 and there is no separation pair in G
4 3-Connectivity Let G=(V,E) be a simple finite graph, n= V, m= E. Def. G is 3-connected n > 3 and there is no separation pair in G Thm (Steinitz 1922). The graphs of convex 3-dimensional polyhedra are exactly the (simple) planar 3-connected graphs.
5 Tait's Conjecture Tait's Conjecture (1884). Every cubic planar 3-connected graph is Hamiltonian. Peter G. Tait
6 Tait's Conjecture Tait's Conjecture (1884). Every cubic planar 3-connected graph is Hamiltonian. every vertex is incident to exactly 3 edges Peter G. Tait
7 Tait's Conjecture Tait's Conjecture (1884). Every cubic planar 3-connected graph is Hamiltonian. every vertex is incident to exactly 3 edges? would imply the 4-color theorem Peter G. Tait
8 Tait's Conjecture Tait's Conjecture (1884). Every cubic planar 3-connected graph is Hamiltonian William T. Tutte Tutte's fragment
9 Tait's Conjecture Tait's Conjecture (1884). Every cubic planar 3-connected graph is Hamiltonian. Tutte's counterexample (1946, 46 vertices) William T. Tutte
10 Tutte's Conjecture planar Tutte's Conjecture (1971). Every cubic bipartite 3-connected graph is Hamiltonian. William T. Tutte
11 Tutte's Conjecture planar Tutte's Conjecture (1971). Every cubic bipartite 3-connected graph is Hamiltonian.? William T. Tutte
12 Tutte's Conjecture planar Tutte's Conjecture (1971). Every cubic bipartite 3-connected graph is Hamiltonian. Joseph D. Horton Counterexamples of Baraev & Faradzhev (1978) and Horton (1982)
13 Barnette's Conjecture Barnette's Conjecture (1969). Every cubic bipartite planar 3-connected graph is Hamiltonian. David W. Barnette
14 Barnette's Conjecture Barnette's Conjecture (1969). Every cubic bipartite planar 3-connected graph is Hamiltonian.?still open... David W. Barnette
15 Barnette's Conjecture Barnette's Conjecture (1969). Every cubic bipartite planar 3-connected graph is Hamiltonian....but on the edge: Thm (Tutte 1956). Every planar 4-connected graph is Hamiltonian.
16 Barnette's Conjecture Barnette's Conjecture (1969). Every cubic bipartite planar 3-connected graph is Hamiltonian....but on the edge: Thm (Tutte 1956). Every planar 4-connected graph is Hamiltonian. Without assuming cubicness, non-hamiltonian graphs exist.
17 Barnette's Conjecture Barnette's Conjecture (1969). Every cubic bipartite planar 3-connected graph is Hamiltonian....but on the edge: Thm (Tutte 1956). Every planar 4-connected graph is Hamiltonian. Without assuming cubicness, non-hamiltonian graphs exist. Kirkman graph
18 Barnette's Conjecture Barnette's Conjecture (1969). Every cubic bipartite planar 3-connected graph is Hamiltonian....but on the edge: Barnette graph Thm (Tutte 1956). Every planar 4-connected graph is Hamiltonian. Without assuming cubicness, non-hamiltonian graphs exist. Kirkman graph
19 Strengthening the Conjecture A Barnette graph is x-y-hamiltonian if for every two edges x and y in a common face f there is a Hamiltonian cycle that contains x but not y. Thm (Kelmans 1986). Barnette's conjecture is true if and only if every Barnette graph is x-y-hamiltonian.
20 Strengthening the Conjecture A Barnette graph is x-y-hamiltonian if for every two edges x and y in a common face f there is a Hamiltonian cycle that contains x but not y. Thm (Kelmans 1986). Barnette's conjecture is true if and only if every Barnette graph is x-y-hamiltonian. Thm (Hertel 2005). Barnette's conjecture is true if and only if every Barnette graph is x-y-z-hamiltonian where xyz is a path of length 3. x z y
21 Strengthening the Conjecture A Barnette graph is x-y-hamiltonian if for every two edges x and y in a common face f there is a Hamiltonian cycle that contains x but not y. Thm (Kelmans 1986). Barnette's conjecture is true if and only if every Barnette graph is x-y-hamiltonian. Thm (Hertel 2005). Barnette's conjecture is true if and only if every Barnette graph is x-y-z-hamiltonian where xyz is a path of length 3. Proof. <=: easy
22 Strengthening the Conjecture A Barnette graph is x-y-hamiltonian if for every two edges x and y in a common face f there is a Hamiltonian cycle that contains x but not y. Thm (Kelmans 1986). Barnette's conjecture is true if and only if every Barnette graph is x-y-hamiltonian. Thm (Hertel 2005). Barnette's conjecture is true if and only if every Barnette graph is x-y-z-hamiltonian where xyz is a path of length 3. Proof. <=: easy =>: x y z
23 Strengthening the Conjecture A Barnette graph is x-y-hamiltonian if for every two edges x and y in a common face f there is a Hamiltonian cycle that contains x but not y. Thm (Kelmans 1986). Barnette's conjecture is true if and only if every Barnette graph is x-y-hamiltonian. Thm (Hertel 2005). Barnette's conjecture is true if and only if every Barnette graph is x-y-z-hamiltonian where xyz is a path of length 3. Proof. <=: easy =>: x y
24 Strengthening the Conjecture A Barnette graph is x-y-hamiltonian if for every two edges x and y in a common face f there is a Hamiltonian cycle that contains x but not y. Thm (Kelmans 1986). Barnette's conjecture is true if and only if every Barnette graph is x-y-hamiltonian. Thm (Hertel 2005). Barnette's conjecture is true if and only if every Barnette graph is x-y-z-hamiltonian where xyz is a path of length 3. Proof. <=: easy =>: x y
25 Strengthening the Conjecture A Barnette graph is x-y-hamiltonian if for every two edges x and y in a common face f there is a Hamiltonian cycle that contains x but not y. Thm (Kelmans 1986). Barnette's conjecture is true if and only if every Barnette graph is x-y-hamiltonian. Thm (Hertel 2005). Barnette's conjecture is true if and only if every Barnette graph is x-y-z-hamiltonian where xyz is a path of length 3. Proof. <=: easy =>: x y
26 Strengthening the Conjecture A Barnette graph is x-y-hamiltonian if for every two edges x and y in a common face f there is a Hamiltonian cycle that contains x but not y. Thm (Kelmans 1986). Barnette's conjecture is true if and only if every Barnette graph is x-y-hamiltonian. Thm (Hertel 2005). Barnette's conjecture is true if and only if every Barnette graph is x-y-z-hamiltonian where xyz is a path of length 3. Proof. <=: easy =>:
27 Strengthening the Conjecture A Barnette graph is x-y-hamiltonian if for every two edges x and y in a common face f there is a Hamiltonian cycle that contains x but not y. Thm (Kelmans 1986). Barnette's conjecture is true if and only if every Barnette graph is x-y-hamiltonian. Thm (Hertel 2005). Barnette's conjecture is true if and only if every Barnette graph is x-y-z-hamiltonian where xyz is a path of length 3. Proof. <=: easy =>: y
28 Dualization & Computer-aided Search Searching for counterexamples: Thm (Brinkmann, McKay 2002). Every Barnette graph with at most 84 vertices is Hamiltonian. Thm (Brinkmann, McKay 2002). Every Barnette graph with at most 60 vertices is x-y-hamiltonian. Even true if x and y are not necessarily in the same face.
29 Weakening the Conjecture Def. A graph is cyclically 4-edge-connected if every cycle-separating edge cut contains at least 4 edges. not cyclically 4-edge-connected cyclically 4-edge-connected
30 Weakening the Conjecture Thm (S.). Barnette's conjecture is true if and only if every cyclically 4- edge-connected Barnette graph is x-y-hamiltonian. Proof. =>: easy.
31 Weakening the Conjecture Thm (S.). Barnette's conjecture is true if and only if every cyclically 4- edge-connected Barnette graph is x-y-hamiltonian. Proof. =>: easy. <=: Take any non-cyclically 4-edge-connected Barnette graph. A B
32 Weakening the Conjecture Thm (S.). Barnette's conjecture is true if and only if every cyclically 4- edge-connected Barnette graph is x-y-hamiltonian. Proof. =>: easy. <=: Take any non-cyclically 4-edge-connected Barnette graph. A B - cubic
33 Weakening the Conjecture Thm (S.). Barnette's conjecture is true if and only if every cyclically 4- edge-connected Barnette graph is x-y-hamiltonian. Proof. =>: easy. <=: Take any non-cyclically 4-edge-connected Barnette graph. A B - cubic - simple
34 Weakening the Conjecture Thm (S.). Barnette's conjecture is true if and only if every cyclically 4- edge-connected Barnette graph is x-y-hamiltonian. Proof. =>: easy. <=: Take any non-cyclically 4-edge-connected Barnette graph. A B - cubic - simple - 3-connected
35 Weakening the Conjecture Thm (S.). Barnette's conjecture is true if and only if every cyclically 4- edge-connected Barnette graph is x-y-hamiltonian. Proof. =>: easy. <=: Take any non-cyclically 4-edge-connected Barnette graph. A B - cubic - simple - 3-connected - bipartite
36 Weakening the Conjecture Thm (S.). Barnette's conjecture is true if and only if every cyclically 4- edge-connected Barnette graph is x-y-hamiltonian. Proof. =>: easy. <=: Take any non-cyclically 4-edge-connected Barnette graph. A B - cubic - simple - 3-connected - bipartite
37 Special Faces Thm (Goodey 1975). Every Barnette graph that contains only faces of degree 4 and 6 (and at most one special face of degree 8) is Hamiltonian.
38 Special Faces Thm (Goodey 1975). Every Barnette graph that contains only faces of degree 4 and 6 (and at most one special face of degree 8) is Hamiltonian. Thm (Feder, Subi 2009). Every Barnette graph that contains only faces of degree 4 and 6 for two of the three face-color classes is Hamiltonian.
39 Special Faces Thm (Goodey 1975). Every Barnette graph that contains only faces of degree 4 and 6 (and at most one special face of degree 8) is Hamiltonian. Thm (Feder, Subi 2009). Every Barnette graph that contains only faces of degree 4 and 6 for two of the three face-color classes is Hamiltonian. Computing a Hamiltonian cycle for every such graph is in P.
40 End of the first talk. Don't run away.
A Survey & Strengthening of Barnette s Conjecture
A Survey & Strengthening of Barnette s Conjecture Alexander Hertel Department of Computer Science University of Toronto April 29th, 2005 Abstract Tait and Tutte made famous conjectures stating that all
Cycles 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.
Decomposition Theorems for Square-free 2-matchings in Bipartite Graphs
Decomposition Theorems for Square-free 2-matchings in Bipartite Graphs Kenjiro Takazawa RIMS, Kyoto University ISMP 2015 Pittsburgh July 13, 2015 1 Overview G = (V,E): Bipartite, Simple M E: Square-free
Nonhamiltonian 3-connected cubic planar graphs
Nonhamiltonian -connected cubic planar graphs R. E. L. Aldred, S. Bau, D. A. Holton Department of Mathematics and Statistics University of Otago P.O. Box 56, Dunedin, New Zealand raldred@maths.otago.ac.nz
A counterexample to the pseudo 2-factor isomorphic graph conjecture
A counterexample to the pseudo 2-factor isomorphic graph conjecture Jan Goedgebeur a,1 arxiv:1412.3350v2 [math.co] 27 May 2015 Abstract a Department of Applied Mathematics, Computer Science & Statistics
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.
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:
Graphs & 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
Matthews-Sumner Conjecture and Equivalences
University of Memphis June 21, 2012 Forbidden Subgraphs Definition A graph G is H-free if G contains no induced copy of the graph H as a subgraph. More generally, we say G is F-free for some family of
Hamiltonicity of maximal planar graphs and planar triangulations
Hamiltonicity of maximal planar graphs and planar triangulations Von der Fakultät für Mathematik, Informatik und aturwissenschaften der Rheinisch-Westfälischen Technischen Hochschule Aachen zur Erlangung
12.1 Chromatic Number
12.1 Chromatic Number History of Graph Colorings Theorem The Four Color Theorem: Any planar graph can be colored in at most four colors. But, it wasn t always the Four Color Theorem... Four Color Conjecture
arxiv: v1 [math.co] 27 Aug 2015
![arxiv: v1 [math.co] 27 Aug 2015](/thumbs/96/128712557.jpg "arxiv: v1 [math.co] 27 Aug 2015")
arxiv:1508.06934v1 [math.co] 27 Aug 2015 TRIANGLE-FREE UNIQUELY 3-EDGE COLORABLE CUBIC GRAPHS SARAH-MARIE BELCASTRO AND RUTH HAAS Abstract. This paper presents infinitely many new examples of triangle-free
Extremal graphs for some problems on cycles in graphs
Extremal graphs for some problems on cycles in graphs Klas Markström Abstract. This paper contain a collection of extremal graphs for somequestionsoncyclesingraphs.thegraphshavebeenfoundby exhaustive computer
Advanced 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
Perfect 1-factorizations
Perfect 1-factorizations Alexander Rosa Department of Mathematics and Statistics, McMaster University Rosa (McMaster) Perfect 1-factorizations 1 / 17 Smolenice 1963 Rosa (McMaster) Perfect 1-factorizations
Tough 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
P 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
Types of triangle and the impact on domination and k-walks
Types of triangle and the impact on domination and k-walks Gunnar Brinkmann Applied Mathematics, Computer Science and Statistics Krijgslaan 8 S9 Ghent University B9 Ghent gunnar.brinkmann@ugent.be Kenta
On 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
Cubic vertices in planar hypohamiltonian graphs
Cubic vertices in planar hypohamiltonian graphs Carol T. ZAMFIRESCU Abstract. Thomassen showed in 1978 that every planar hypohamiltonian graph contains a cubic vertex. Equivalently, a planar graph with
Hamiltonian 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
Uniquely Hamiltonian Graphs
Uniquely Hamiltonian Graphs Benedikt Klocker Algorithms and Complexity Group Institute of Computer Graphics and Algorithms TU Wien Retreat Talk Uniquely Hamiltonian Graphs Benedikt Klocker 2 Basic Definitions
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.
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
2 Definitions We begin by reviewing the construction and some basic properties of squeezed balls and spheres [5, 7]. For an integer n 1, let [n] denot
![2 Definitions We begin by reviewing the construction and some basic properties of squeezed balls and spheres [5, 7]. For an integer n 1, let [n] denot](/thumbs/89/98546623.jpg "2 Definitions We begin by reviewing the construction and some basic properties of squeezed balls and spheres [5, 7]. For an integer n 1, let [n] denot")
Squeezed 2-Spheres and 3-Spheres are Hamiltonian Robert L. Hebble Carl W. Lee December 4, 2000 1 Introduction A (convex) d-polytope is said to be Hamiltonian if its edge-graph is. It is well-known that
Cycles in 4-Connected Planar Graphs
Cycles in 4-Connected Planar Graphs Guantao Chen Department of Mathematics & Statistics Georgia State University Atlanta, GA 30303 matgcc@panther.gsu.edu Genghua Fan Institute of Systems Science Chinese
Graph Theory and Interconnection Networks
Graph Theory and Interconnection Networks Lih-Hsing Hsu and Cheng-Kuan Lin @ CRC Press Taylor & Francis Group Boca Raton London New York CRC Press is an imprint of the Taylor & Francis Group, an informa
1-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
Siegel s theorem, edge coloring, and a holant dichotomy
Siegel s theorem, edge coloring, and a holant dichotomy Tyson Williams (University of Wisconsin-Madison) Joint with: Jin-Yi Cai and Heng Guo (University of Wisconsin-Madison) Appeared at FOCS 2014 1 /
Genetic Theory of Cubic Graphs
Hamiltonian Cycle, Traveling Salesman and Related Optimization Problems Workshop, 2012 Joint work with: Vladimir Ejov, Jerzy Filar, Michael Haythorpe Earlier works containing ideas that can be seen as
Snarks with total chromatic number 5
Snarks with total chromatic number 5 Gunnar Brinkmann, Myriam Preissmann, Diana Sasaki To cite this version: Gunnar Brinkmann, Myriam Preissmann, Diana Sasaki. Snarks with total chromatic number 5. Discrete
On the connectivity of extremal Ramsey graphs
On the connectivity of extremal Ramsey graphs Andrew Beveridge and Oleg Pikhurko Department of Mathematical Sciences Carnegie Mellon University Pittsburgh, PA 15213 Abstract An (r, b)-graph is a graph
Perfect 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
Advanced Combinatorial Optimization September 24, Lecture 5
18.438 Advanced Combinatorial Optimization September 24, 2009 Lecturer: Michel X. Goemans Lecture 5 Scribe: Yehua Wei In this lecture, we establish the connection between nowhere-zero (nwz) k-flow and
M-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
arxiv: v1 [math.co] 21 Jan 2016
![arxiv: v1 [math.co] 21 Jan 2016](/thumbs/95/124148803.jpg "arxiv: 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
Cycle 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;
Long Cycles in 3-Connected Graphs
Long Cycles in 3-Connected Graphs Guantao Chen Department of Mathematics & Statistics Georgia State University Atlanta, GA 30303 Xingxing Yu School of Mathematics Georgia Institute of Technology Atlanta,
P 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
An approximate version of Hadwiger s conjecture for claw-free graphs
An approximate version of Hadwiger s conjecture for claw-free graphs Maria Chudnovsky Columbia University, New York, NY 10027, USA and Alexandra Ovetsky Fradkin Princeton University, Princeton, NJ 08544,
Colourings 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
Infinite circuits in infinite graphs
Infinite circuits in infinite graphs Henning Bruhn Universität Hamburg R. Diestel, A. Georgakopoulos, D. Kühn, P. Sprüssel, M. Stein Henning Bruhn (U Hamburg) Infinite circuits Haifa 08 1 / 25 Locally
Hypohamiltonian and Almost Hypohamiltonian Graphs
Hypohamiltonian and Almost Hypohamiltonian Graphs Carol T. Zamfirescu Promotor: prof. dr. Gunnar Brinkmann Vakgroep Toegepaste Wiskunde, Informatica en Statistiek Faculteit Wetenschappen Proefschrift ingediend
SAT, Coloring, Hamiltonian Cycle, TSP
1 SAT, Coloring, Hamiltonian Cycle, TSP Slides by Carl Kingsford Apr. 28, 2014 Sects. 8.2, 8.7, 8.5 2 Boolean Formulas Boolean Formulas: Variables: x 1, x 2, x 3 (can be either true or false) Terms: t
RECAP: Extremal problems Examples
RECAP: Extremal problems Examples Proposition 1. If G is an n-vertex graph with at most n edges then G is disconnected. A Question you always have to ask: Can we improve on this proposition? Answer. NO!
Computational Intractability 2010/4/15. Lecture 2
Computational Intractability 2010/4/15 Professor: David Avis Lecture 2 Scribe:Naoki Hatta 1 P and NP 1.1 Definition of P and NP Decision problem it requires yes/no answer. Example: X is a set of strings.
Hamiltonian Cycles in Sparse Graphs
Hamiltonian Cycles in Sparse Graphs By Alexander Hertel A Thesis Submitted in Conformity With the Requirements For the Degree of Master of Science Graduate Department of Computer Science University of
Cycle Double Cover Conjecture
Cycle Double Cover Conjecture Paul Clarke St. Paul's College Raheny January 5 th 2014 Abstract In this paper, a proof of the cycle double cover conjecture is presented. The cycle double cover conjecture
Cubic 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
On a conjecture concerning the Petersen graph
On a conjecture concerning the Petersen graph Donald Nelson Department of Mathematical Sciences Middle Tennessee State University Murfreesboro, TN 37132, USA dnelson@mtsu.edu Michael D. Plummer Department
Hamilton 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
Independent 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,
SPECIAL T K 5 IN GRAPHS CONTAINING K 4
SPECIAL T K 5 IN GRAPHS CONTAINING K 4 A Thesis Presented to The Academic Faculty by Dawei He In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in Mathematics School of Mathematics
Advanced Combinatorial Optimization Updated February 18, Lecture 5. Lecturer: Michel X. Goemans Scribe: Yehua Wei (2009)
18.438 Advanced Combinatorial Optimization Updated February 18, 2012. Lecture 5 Lecturer: Michel X. Goemans Scribe: Yehua Wei (2009) In this lecture, we establish the connection between nowhere-zero k-flows
Siegel s Theorem, Edge Coloring, and a Holant Dichotomy
Siegel s Theorem, Edge Coloring, and a Holant Dichotomy Jin-Yi Cai (University of Wisconsin-Madison) Joint with: Heng Guo and Tyson Williams (University of Wisconsin-Madison) 1 / 117 2 / 117 Theorem (Siegel
c 2015 Society for Industrial and Applied Mathematics
SIAM J. DISCRETE MATH. Vol. 29, No. 2, pp. 915 939 c 2015 Society for Industrial and Applied Mathematics TSP TOURS IN CUBIC GRAPHS: BEYOND 4/3 JOSÉ CORREA, OMAR LARRÉ, AND JOSÉ A. SOTO Abstract. After
CORES, 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
Decomposing plane cubic graphs
Decomposing plane cubic graphs Kenta Ozeki and Dong Ye Abstract It was conjectured by Hoffmann-Ostenhof that the edge set of every cubic graph can be decomposed into a spanning tree, a matching and a family
Malaya 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
Planar 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
Mathematical aspects of fullerenes
Mathematical aspects of fullerenes Vesna Andova, František Kardoš, Riste Škrekovski To cite this version: Vesna Andova, František Kardoš, Riste Škrekovski. Mathematical aspects of fullerenes. Ars Mathematica
Even cycle decompositions of 4-regular graphs and line graphs
Even cycle decompositions of 4-regular graphs and line graphs Klas Markström Abstract. Anevencycledecompositionofagraphisapartitionofitsedgeintoeven cycles. We first give some results on the existence
Semiregular 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
EXCLUDING MINORS IN NONPLANAR GRAPHS OFGIRTHATLEASTFIVE. Robin Thomas 1 and. Jan McDonald Thomson 2
EXCLUDING MINORS IN NONPLANAR GRAPHS OFGIRTHATLEASTFIVE Robin Thomas 1 thomas@math.gatech.edu and Jan McDonald Thomson 2 thomson@math.gatech.edu School of Mathematics Georgia Institute of Technology Atlanta,
Group flow, complex flow, unit vector flow, and the (2+)-flow conjecture
Downloaded from orbit.dtu.dk on: Jan 14, 2019 Group flow, complex flow, unit vector flow, and the (2+)-flow conjecture Thomassen, Carsten Published in: Journal of Combinatorial Theory. Series B Link to
Colourings 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
A Recoverable Annular Network with Unrecoverable Dual
A Recoverable Annular Network with Unrecoverable Dual David Jekel August 18, 213 Recovering the Network For definitions, see [1]. I consider the network Γ shown in Figure 1 and its dual Γ shown in Figure
Semiregular 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
MATH 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
Odd 2 factored snarks
arxiv:1210.8101v2 [math.co] 21 Jan 2013 Odd 2 factored snarks M. Abreu, Dipartimento di Matematica, Informatica ed Economia, Università della Basilicata, C. da Macchia Romana, 85100 Potenza, Italy. e-mail:
MATH 363: Discrete Mathematics
MATH 363: Discrete Mathematics Learning Objectives by topic The levels of learning for this class are classified as follows. 1. Basic Knowledge: To recall and memorize - Assess by direct questions. The
Ringing the Changes II
Ringing the Changes II 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
CYCLICALLY FIVE CONNECTED CUBIC GRAPHS. Neil Robertson 1 Department of Mathematics Ohio State University 231 W. 18th Ave. Columbus, Ohio 43210, USA
CYCLICALLY FIVE CONNECTED CUBIC GRAPHS Neil Robertson 1 Department of Mathematics Ohio State University 231 W. 18th Ave. Columbus, Ohio 43210, USA P. D. Seymour 2 Department of Mathematics Princeton University
Some Results on Paths and Cycles in Claw-Free Graphs
Some Results on Paths and Cycles in Claw-Free Graphs BING WEI Department of Mathematics University of Mississippi 1 1. Basic Concepts A graph G is called claw-free if it has no induced subgraph isomorphic
The Matching Polytope: General graphs
8.433 Combinatorial Optimization The Matching Polytope: General graphs September 8 Lecturer: Santosh Vempala A matching M corresponds to a vector x M = (0, 0,,, 0...0) where x M e is iff e M and 0 if e
Polynomial-time reductions. We have seen several reductions:
Polynomial-time reductions We have seen several reductions: Polynomial-time reductions Informal explanation of reductions: We have two problems, X and Y. Suppose we have a black-box solving problem X in
2 hours THE UNIVERSITY OF MANCHESTER. 6 June :45 11:45
2 hours THE UNIVERSITY OF MANCHESTER DISCRETE MATHEMATICS 6 June 2016 9:45 11:45 Answer ALL THREE questions in Section A (30 marks in total) and TWO of the THREE questions in Section B (50 marks in total).
Planar 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
HAMILTONIAN PROPERTIES OF TRIANGULAR GRID GRAPHS. 1. Introduction
HAMILTONIAN PROPERTIES OF TRIANGULAR GRID GRAPHS VALERY S. GORDON, YURY L. ORLOVICH, FRANK WERNER Abstract. A triangular grid graph is a finite induced subgraph of the infinite graph associated with the
Excluded t-factors in Bipartite Graphs:
Excluded t-factors in Bipartite Graphs: A nified Framework for Nonbipartite Matchings and Restricted -matchings Blossom and Subtour Elimination Constraints Kenjiro Takazawa Hosei niversity, Japan IPCO017
Properly colored Hamilton cycles in edge colored complete graphs
Properly colored Hamilton cycles in edge colored complete graphs N. Alon G. Gutin Dedicated to the memory of Paul Erdős Abstract It is shown that for every ɛ > 0 and n > n 0 (ɛ), any complete graph K on
arxiv: v1 [math.co] 22 Jan 2018
![arxiv: v1 [math.co] 22 Jan 2018](/thumbs/86/94250040.jpg "arxiv: v1 [math.co] 22 Jan 2018")
arxiv:1801.07025v1 [math.co] 22 Jan 2018 Spanning trees without adjacent vertices of degree 2 Kasper Szabo Lyngsie, Martin Merker Abstract Albertson, Berman, Hutchinson, and Thomassen showed in 1990 that
Spanning 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
FRACTIONAL PACKING OF T-JOINS. 1. Introduction
FRACTIONAL PACKING OF T-JOINS FRANCISCO BARAHONA Abstract Given a graph with nonnegative capacities on its edges, it is well known that the capacity of a minimum T -cut is equal to the value of a maximum
A zero-free interval for chromatic polynomials of graphs with 3-leaf spanning trees
Downloaded from orbit.dtu.dk on: Sep 02, 2018 A zero-free interval for chromatic polynomials of graphs with 3-leaf spanning trees Perrett, Thomas Published in: Discrete Mathematics Link to article, DOI:
Combinatorics and Optimization 442/642, Fall 2012
Compact course notes Combinatorics and Optimization 44/64, Fall 0 Graph Theory Contents Professor: J. Geelen transcribed by: J. Lazovskis University of Waterloo December 6, 0 0. Foundations..............................................
Improved Approximations for Cubic Bipartite and Cubic TSP
Improved Approximations for Cubic Bipartite and Cubic TSP Anke van Zuylen Department of Mathematics The College of William and Mary, Williamsburg, VA, 23185, USA anke@wm.edu Abstract We show improved approximation
Finding many edge-disjoint Hamiltonian cycles in dense graphs
Finding many edge-disjoint Hamiltonian cycles in dense graphs Stephen G. Hartke Department of Mathematics University of Nebraska Lincoln www.math.unl.edu/ shartke2 hartke@math.unl.edu Joint work with Tyler
arxiv: v1 [math.co] 3 Aug 2009
![arxiv: v1 [math.co] 3 Aug 2009](/thumbs/93/114197455.jpg "arxiv: v1 [math.co] 3 Aug 2009")
GRAPHS WHOSE FLOW POLYNOMIALS HAVE ONLY INTEGRAL ROOTS arxiv:0908.0181v1 [math.co] 3 Aug 009 JOSEPH P.S. KUNG AND GORDON F. ROYLE Abstract. We show if the flow polynomial of a bridgeless graph G has only
AN INTRODUCTION TO CHROMATIC POLYNOMIALS
BACKGROUND POLYNOMIALS COMPLEX ZEROS AN INTRODUCTION TO CHROMATIC POLYNOMIALS Gordon Royle School of Mathematics & Statistics University of Western Australia Junior Mathematics Seminar, UWA September 2011
Consecutive ones Block for Symmetric Matrices
Consecutive ones Block for Symmetric Matrices Rui Wang, FrancisCM Lau Department of Computer Science and Information Systems The University of Hong Kong, Hong Kong, PR China Abstract We show that a cubic
Random regular graphs. Nick Wormald University of Waterloo
Random regular graphs Nick Wormald University of Waterloo LECTURE 5 Random regular graphs What are they (models) Typical and powerful results Handles for analysis One or two general techniques Some open
The Probabilistic Method
The Probabilistic Method In Graph Theory Ehssan Khanmohammadi Department of Mathematics The Pennsylvania State University February 25, 2010 What do we mean by the probabilistic method? Why use this method?
Percolation on random triangulations
Percolation on random triangulations Olivier Bernardi (MIT) Joint work with Grégory Miermont (Université Paris-Sud) Nicolas Curien (École Normale Supérieure) MSRI, January 2012 Model and motivations Planar
arxiv: v2 [math.co] 7 Jan 2016
![arxiv: v2 [math.co] 7 Jan 2016](/thumbs/92/110891379.jpg "arxiv: 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,
Algorithms 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
Extremal Graphs Having No Stable Cutsets
Extremal Graphs Having No Stable Cutsets Van Bang Le Institut für Informatik Universität Rostock Rostock, Germany le@informatik.uni-rostock.de Florian Pfender Department of Mathematics and Statistics University
GRAPH MINORS AND HADWIGER S CONJECTURE
GRAPH MINORS AND HADWIGER S CONJECTURE DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Eliade
On the Path-width of Planar Graphs
On the Path-width of Planar Graphs Omid Amini 1,2 Florian Huc 1 Stéphane Pérennes 1 FirstName.LastName@sophia.inria.fr Abstract In this paper, we present a result concerning the relation between the path-with
Compatible Hamilton cycles in Dirac graphs
Compatible Hamilton cycles in Dirac graphs Michael Krivelevich Choongbum Lee Benny Sudakov Abstract A graph is Hamiltonian if it contains a cycle passing through every vertex exactly once. A celebrated
A shorter proof of Thomassen s theorem on Tutte paths in plane graphs
A shorter proof of Thomassen s theorem on Tutte paths in plane graphs Kenta Ozeki National Institute of Informatics, 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo 101-8430, Japan and JST, ERATO, Kawarabayashi
A study of necessary and sufficient conditions for vertex transitive graphs to be Hamiltonian
A study of necessary and sufficient conditions for vertex transitive graphs to be Hamiltonian Annelies Heus Master s thesis under supervision of dr. D. Gijswijt University of Amsterdam, Faculty of Science