Minimum Size of Bipartite-Saturated Graphs
|
|
- Justina Richards
- 5 years ago
- Views:
Transcription
1 Middlebury College joint work with Oleg Pikhurko (Carnegie Mellon University) July, 007 British Combinatorics Conference (Reading, UK)
2 Definitions History Definition A graph G is F-saturated if F G, F G + e for any e E(G). Problem Determine the minimum number of edges,sat(n,f), of an F-saturated graph.
3 Definitions History Theorem (Erdős, Hajnal, Moon ) ( ) t sat(n,k t ) = (t )(n 1) Furthermore, the only K t -saturated graph with this many edges is K t + K n t+. K t K n t+
4 Both small and large Luczak Wheel Theorem (Ollmann - 7, Tuza - 86, Fisher, Fraughnaugh, Langley - 97) sat(n,c 4 ) = 3n 5, n 5
5 Both small and large Luczak Wheel Theorem (Ollmann - 7, Tuza - 86, Fisher, Fraughnaugh, Langley - 97) sat(n,c 4 ) = 3n 5, n 5
6 Both small and large Luczak Wheel Theorem (Y.C.Chen, 07+) sat(n,c 5 ) = 10n 10,n 1 7
7 Hamiltonian Both small and large Luczak Wheel Theorem sat(n,c n ) = 3n + 1,n 53 Bondy ( 7) showed the lower bound. Clark, Entringer, Crane and Shapiro ( 83-86) gave upper bound based on Isaacs flower snarks (girth 5, 6). L. Stacho ( 96) gave further constructions based on the Coxeter graph (girth 7). Problem (Horák, Širáň - 86) Is there a maximally non-hamiltonian graph of girth at least 8 (that meets this bound)?
8 Both small and large Luczak Wheel Conjecture (Bollobás - 78) n + c 1 n l sat(n,c l ) n + c n l Theorem (Barefoot, Clark, Entringer, Porter, Székely, Tuza - 96) (1 + 1 l + 8 )n sat(n,c l)
9 Both small and large Luczak Wheel Theorem (Barefoot et al. - 96) sat(n,c l ) (1 + 6 l 3 )n + O(l ) for l odd, l 9 sat(n,c l ) (1 + 4 l )n + O(l3 ) for l even, l 14
10 Both small and large Luczak Wheel Theorem (Barefoot et al. - 96) [Gould, Luczak, S. - 06] sat(n,c l ) ( l 3 l 3 )n + 4 for l odd, l 9, l 17,n 7l sat(n,c l ) ( l )n + 5l 4 for l even l 14, l 10,n 3l
11 Both small and large Luczak Wheel The Even Luczak Wheel, l = k + 10 Ê Ñ Ò «Ö ½ Ö ¾ «ÀÙ «Ð Ò Ö ½ Ò ½ Ã ËÔÓ ËÔÓ ¹ÒÙØ
12 Both small and large Luczak Wheel The Even Luczak Wheel, l = k + 10 Ê Ñ Ò «Ö ½ Ö ¾ «ÀÙ «Ð Ò Ö ½ Ò ½ Ã ËÔÓ ËÔÓ ¹ÒÙØ
13 Both small and large Luczak Wheel The Even Luczak Wheel, l = k + 10 Ê Ñ Ò «Ö ½ Ö ¾ «ÀÙ «Ð Ò Ö ½ Ò ½ Ã ËÔÓ ËÔÓ ¹ÒÙØ
14 Other Subgraphs Other values of sat(n,f) known for: matchings (Mader - 73), paths and stars (Kászonyi and Tuza - 86), hamiltonian path, P n (Frick and Singleton, 05; Dudek, Katona, Wojda - 06) sat(n,p n ) = 3n,n 54 longest path = detour(beineke, Dunbar, Frick, 05)
15 Difficulties and Hereditary Properties Lacking Quote from Erdős, Hajnal and Moon: One of the difficulties of proving these conjectures may be that the obvious extremal graphs are certainly not unique, which fact may make an induction proof difficult. sat(n,f) sat(n + 1,F) F 1 F sat(n, F 1 ) sat(n, F ) F F sat(n,f ) sat(n,f)
16 Best known upper bound Theorem (Kászonyi and Tuza) Let F be a graph. Set u := u(f) = V (F) α(f) 1 s := s(f) = min{e(f ) : F F,α(F ) = α(f), V (F ) = α(f)+1}. Then sat(n,f) (u + s 1 u(s + u) )n. They considered a clique on u vertices joined to an (s 1)-regular graph.
17 Best Known Lower Bound???? Problem For an arbitrary graph F, determine a non-trivial lower bound on sat(n,f).
18 Saturation for Bipartite Graphs,,s / / (s 1) regular /
19 Saturation for Bipartite Graphs,,s / / (s 1) regular /
20 Saturation for Bipartite Graphs,,s / / (s 1) regular / Yields an improvement of a constant times s over the KT bound.
21 Theorem (O.Pikhurko, S.) There is a constant C such that for all n 5 we have n Cn 3/4 sat(n,k,3 ) n 3.
22 Theorem (O.Pikhurko, S.) There is a constant C such that for all n 5 we have n Cn 3/4 sat(n,k,3 ) n 3.
23 Theorem (O.Pikhurko, S.) There is a constant C such that for all n 5 we have n Cn 3/4 sat(n,k,3 ) n 3.
24 Proof of Lower Bound Let G be a K,3 -saturated graph.
25 Proof of Lower Bound Let G be a K,3 -saturated graph. If δ(g) 4, then E(G) n and we are done.
26 Proof of Lower Bound Let G be a K,3 -saturated graph. If δ(g) = 1 then,
27 Proof of Lower Bound If δ(g) = 1 then,
28 Proof of Lower Bound If δ(g) = 1 then, and so E(G) n 3.
29 Proof of Lower Bound Otherwise, δ(g) 3, pick vertex of minimum degree and consider breadth-first search tree. V1 V V3
30 Proof of Lower Bound Otherwise, δ(g) 3, pick vertex of minimum degree and consider breadth-first search tree. V1 V V3 Tree has n 1 edges, we must find n Cn 3/4 more edges.
31 Divide and Conquer V1 V V3 Y0 Y Y 1
32 Divide and Conquer V1 V V3 Y0 Y Y 1
33 Hit em where they re weakest V1 V V3 Y0 Y Y 1 Y 0 has at most one component which is a tree. Pick up an extra V 3 1 edges.
34 More Division and More Conquering V1 V X 0 X 1 X V3 Pick up extra V #(trees in X 0 ) edges.
35 More Division and More Conquering V1 V X 0 X X 1 V3
36 More Hitting Weak Spots V1 V X 0 X X 1 V3 Trees in X 0 are connected via a path of length at most three through V 3.
37 More Hitting Weak Spots V1 V X 0 X 1 X small degree vertices large degree vertices V3 Small degree vertices can only serve so many trees of X 0. So, sum of large degree vertices is large.
38 More Hitting Weak Spots V1 V X 0 X 1 X small degree vertices large degree vertices V3 Small degree vertices can only serve so many trees of X 0. So, sum of large degree vertices is large. This allows us to add #(trees in X 0 ) O(n 3/4 ) edges to the count. Completes proof.
39 Open problems Problem Determine an exact result for sat(n,k,3 ). Problem Determine the asymptotic for sat(n,k 3,3 ). Talk and results are available online at: jschmitt/
A dual to the Turán problem
Middlebury College joint work with Ron Gould (Emory University) Tomasz Luczak (Adam Mickiewicz University and Emory University) Oleg Pikhurko (Carnegie Mellon University) October 008 Dartmouth College
More informationA Survey of Minimum Saturated Graphs
A Survey of Minimum Saturated Graphs Jill R. Faudree Department of Mathematics and Statistics University of Alaska at Fairbanks Fairbanks, AK 99775-6660 jfaudree@alaska.edu Ralph J. Faudree Department
More informationThe Saturation Function of Complete Partite Graphs
The Saturation Function of Complete Partite Graphs Tom Bohman 1,3 Maria Fonoberova Oleg Pikhurko 1,4 August 30, 010 Abstract A graph G is called F -saturated if it is F -free but the addition of any missing
More informationDiscrete Mathematics. The edge spectrum of the saturation number for small paths
Discrete Mathematics 31 (01) 68 689 Contents lists available at SciVerse ScienceDirect Discrete Mathematics journal homepage: www.elsevier.com/locate/disc The edge spectrum of the saturation number for
More informationSATURATION SPECTRUM OF PATHS AND STARS
1 Discussiones Mathematicae Graph Theory xx (xxxx 1 9 3 SATURATION SPECTRUM OF PATHS AND STARS 4 5 6 7 8 9 10 11 1 13 14 15 16 17 18 Jill Faudree Department of Mathematics and Statistics University of
More informationGraph Packing - Conjectures and Results
Graph Packing p.1/23 Graph Packing - Conjectures and Results Hemanshu Kaul kaul@math.iit.edu www.math.iit.edu/ kaul. Illinois Institute of Technology Graph Packing p.2/23 Introduction Let G 1 = (V 1,E
More informationOn 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
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 informationCycles with consecutive odd lengths
Cycles with consecutive odd lengths arxiv:1410.0430v1 [math.co] 2 Oct 2014 Jie Ma Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213. Abstract It is proved that there
More informationOn the Regularity Method
On the Regularity Method Gábor N. Sárközy 1 Worcester Polytechnic Institute USA 2 Computer and Automation Research Institute of the Hungarian Academy of Sciences Budapest, Hungary Co-authors: P. Dorbec,
More informationA NOTE ON THE TURÁN FUNCTION OF EVEN CYCLES
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000 000 S 0002-9939(XX)0000-0 A NOTE ON THE TURÁN FUNCTION OF EVEN CYCLES OLEG PIKHURKO Abstract. The Turán function ex(n, F
More informationNew Results on Graph Packing
Graph Packing p.1/18 New Results on Graph Packing Hemanshu Kaul hkaul@math.uiuc.edu www.math.uiuc.edu/ hkaul/. University of Illinois at Urbana-Champaign Graph Packing p.2/18 Introduction Let G 1 = (V
More informationSaturation in the Hypercube and Bootstrap Percolation
Saturation in the Hypercube and Bootstrap Percolation Natasha Morrison, Jonathan A. Noel, and Alex Scott Mathematical Institute, University of Oxford, Woodstock Road, Oxford OX2 6GG, UK. {morrison,noel,scott}@maths.ox.ac.uk
More informationOn subgraphs of large girth
1/34 On subgraphs of large girth In Honor of the 50th Birthday of Thomas Vojtěch Rödl rodl@mathcs.emory.edu joint work with Domingos Dellamonica May, 2012 2/34 3/34 4/34 5/34 6/34 7/34 ARRANGEABILITY AND
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 informationBalanced bipartitions of graphs
2010.7 - Dedicated to Professor Feng Tian on the occasion of his 70th birthday Balanced bipartitions of graphs Baogang Xu School of Mathematical Science, Nanjing Normal University baogxu@njnu.edu.cn or
More informationLarge topological cliques in graphs without a 4-cycle
Large topological cliques in graphs without a 4-cycle Daniela Kühn Deryk Osthus Abstract Mader asked whether every C 4 -free graph G contains a subdivision of a complete graph whose order is at least linear
More informationOn the number of cycles in a graph with restricted cycle lengths
On the number of cycles in a graph with restricted cycle lengths Dániel Gerbner, Balázs Keszegh, Cory Palmer, Balázs Patkós arxiv:1610.03476v1 [math.co] 11 Oct 2016 October 12, 2016 Abstract Let L be a
More informationCharacterizing extremal limits
Characterizing extremal limits Oleg Pikhurko University of Warwick ICERM, 11 February 2015 Rademacher Problem g(n, m) := min{#k 3 (G) : v(g) = n, e(g) = m} Mantel 1906, Turán 41: max{m : g(n, m) = 0} =
More informationA note on balanced bipartitions
A note on balanced bipartitions Baogang Xu a,, Juan Yan a,b a School of Mathematics and Computer Science Nanjing Normal University, 1 Ninghai Road, Nanjing, 10097, China b College of Mathematics and System
More informationBounds for pairs in partitions of graphs
Bounds for pairs in partitions of graphs Jie Ma Xingxing Yu School of Mathematics Georgia Institute of Technology Atlanta, GA 30332-0160, USA Abstract In this paper we study the following problem of Bollobás
More informationApplications of Eigenvalues in Extremal Graph Theory
Applications of Eigenvalues in Extremal Graph Theory Olivia Simpson March 14, 201 Abstract In a 2007 paper, Vladimir Nikiforov extends the results of an earlier spectral condition on triangles in graphs.
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 informationInduced Turán numbers
Induced Turán numbers Michael Tait Carnegie Mellon University mtait@cmu.edu Atlanta Lecture Series XVIII Emory University October 22, 2016 Michael Tait (CMU) October 22, 2016 1 / 25 Michael Tait (CMU)
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 informationSaturation numbers for Ramsey-minimal graphs
Saturation numbers for Ramsey-minimal graphs Martin Rolek and Zi-Xia Song Department of Mathematics University of Central Florida Orlando, FL 3816 August 17, 017 Abstract Given graphs H 1,..., H t, a graph
More informationarxiv:math/ v1 [math.co] 17 Apr 2002
arxiv:math/0204222v1 [math.co] 17 Apr 2002 On Arithmetic Progressions of Cycle Lengths in Graphs Jacques Verstraëte Department of Pure Mathematics and Mathematical Statistics Centre for Mathematical Sciences
More informationCycle lengths in sparse graphs
Cycle lengths in sparse graphs Benny Sudakov Jacques Verstraëte Abstract Let C(G) denote the set of lengths of cycles in a graph G. In the first part of this paper, we study the minimum possible value
More informationKatarzyna Mieczkowska
Katarzyna Mieczkowska Uniwersytet A. Mickiewicza w Poznaniu Erdős conjecture on matchings in hypergraphs Praca semestralna nr 1 (semestr letni 010/11 Opiekun pracy: Tomasz Łuczak ERDŐS CONJECTURE ON MATCHINGS
More informationRamsey Unsaturated and Saturated Graphs
Ramsey Unsaturated and Saturated Graphs P Balister J Lehel RH Schelp March 20, 2005 Abstract A graph is Ramsey unsaturated if there exists a proper supergraph of the same order with the same Ramsey number,
More informationDisjoint Subgraphs in Sparse Graphs 1
Disjoint Subgraphs in Sparse Graphs 1 Jacques Verstraëte Department of Pure Mathematics and Mathematical Statistics Centre for Mathematical Sciences Wilberforce Road Cambridge CB3 OWB, UK jbav2@dpmms.cam.ac.uk
More informationDecompositions of graphs into cycles with chords
Decompositions of graphs into cycles with chords Paul Balister Hao Li Richard Schelp May 22, 2017 In memory of Dick Schelp, who passed away shortly after the submission of this paper Abstract We show that
More informationIndependent Transversals in r-partite Graphs
Independent Transversals in r-partite Graphs Raphael Yuster Department of Mathematics Raymond and Beverly Sackler Faculty of Exact Sciences Tel Aviv University, Tel Aviv, Israel Abstract Let G(r, n) denote
More informationBounds on the generalised acyclic chromatic numbers of bounded degree graphs
Bounds on the generalised acyclic chromatic numbers of bounded degree graphs Catherine Greenhill 1, Oleg Pikhurko 2 1 School of Mathematics, The University of New South Wales, Sydney NSW Australia 2052,
More informationThe Rainbow Turán Problem for Even Cycles
The Rainbow Turán Problem for Even Cycles Shagnik Das University of California, Los Angeles Aug 20, 2012 Joint work with Choongbum Lee and Benny Sudakov Plan 1 Historical Background Turán Problems Colouring
More informationQuadruple Systems with Independent Neighborhoods
Quadruple Systems with Independent Neighborhoods Zoltan Füredi Dhruv Mubayi Oleg Pikhurko Abstract A -graph is odd if its vertex set can be partitioned into two sets so that every edge intersects both
More informationCycle Spectra of Hamiltonian Graphs
Cycle Spectra of Hamiltonian Graphs Kevin G. Milans, Dieter Rautenbach, Friedrich Regen, and Douglas B. West July, 0 Abstract We prove that every graph consisting of a spanning cycle plus p chords has
More informationProper connection number and 2-proper connection number of a graph
Proper connection number and 2-proper connection number of a graph arxiv:1507.01426v2 [math.co] 10 Jul 2015 Fei Huang, Xueliang Li, Shujing Wang Center for Combinatorics and LPMC-TJKLC Nankai University,
More informationVertex colorings of graphs without short odd cycles
Vertex colorings of graphs without short odd cycles Andrzej Dudek and Reshma Ramadurai Department of Mathematical Sciences Carnegie Mellon University Pittsburgh, PA 1513, USA {adudek,rramadur}@andrew.cmu.edu
More informationT i t l e o f t h e w o r k : L a M a r e a Y o k o h a m a. A r t i s t : M a r i a n o P e n s o t t i ( P l a y w r i g h t, D i r e c t o r )
v e r. E N G O u t l i n e T i t l e o f t h e w o r k : L a M a r e a Y o k o h a m a A r t i s t : M a r i a n o P e n s o t t i ( P l a y w r i g h t, D i r e c t o r ) C o n t e n t s : T h i s w o
More informationPacking nearly optimal Ramsey R(3, t) graphs
Packing nearly optimal Ramsey R(3, t) graphs He Guo Joint work with Lutz Warnke Context of this talk Ramsey number R(s, t) R(s, t) := minimum n N such that every red/blue edge-coloring of complete n-vertex
More informationThe Lopsided Lovász Local Lemma
Joint work with Linyuan Lu and László Székely Georgia Southern University April 27, 2013 The lopsided Lovász local lemma can establish the existence of objects satisfying several weakly correlated conditions
More informationAdvanced Topics in Discrete Math: Graph Theory Fall 2010
21-801 Advanced Topics in Discrete Math: Graph Theory Fall 2010 Prof. Andrzej Dudek notes by Brendan Sullivan October 18, 2010 Contents 0 Introduction 1 1 Matchings 1 1.1 Matchings in Bipartite Graphs...................................
More informationThe Generalised Randić Index of Trees
The Generalised Randić Index of Trees Paul Balister Béla Bollobás Stefanie Gerke December 10, 2006 Abstract The Generalised Randić index R α (T ) of a tree T is the sum over the edges uv of T of (d(u)d(v))
More informationarxiv: v1 [math.co] 2 Dec 2013
What is Ramsey-equivalent to a clique? Jacob Fox Andrey Grinshpun Anita Liebenau Yury Person Tibor Szabó arxiv:1312.0299v1 [math.co] 2 Dec 2013 November 4, 2018 Abstract A graph G is Ramsey for H if every
More informationOn the Pósa-Seymour Conjecture
On the Pósa-Seymour Conjecture János Komlós, 1 Gábor N. Sárközy, 2 and Endre Szemerédi 3 1 DEPT. OF MATHEMATICS, RUTGERS UNIVERSITY, NEW BRUNSWICK, NJ 08903 2 DEPT. OF COMPUTER SCIENCE, WORCESTER POLYTECHNIC
More informationThe Turán number of sparse spanning graphs
The Turán number of sparse spanning graphs Noga Alon Raphael Yuster Abstract For a graph H, the extremal number ex(n, H) is the maximum number of edges in a graph of order n not containing a subgraph isomorphic
More informationAdding random edges to create the square of a Hamilton cycle
Adding random edges to create the square of a Hamilton cycle Patrick Bennett Andrzej Dudek Alan Frieze October 7, 2017 Abstract We consider how many random edges need to be added to a graph of order n
More informationA taste of perfect graphs
A taste of perfect graphs Remark Determining the chromatic number of a graph is a hard problem, in general, and it is even hard to get good lower bounds on χ(g). An obvious lower bound we have seen before
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 informationInduced Saturation Number
Graduate Theses and Dissertations Iowa State University Capstones, Theses and Dissertations 2012 Induced Saturation Number Jason James Smith Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/etd
More informationCutwidth and degeneracy of graphs
Cutwidth and degeneracy of graphs Benoit Kloeckner To cite this version: Benoit Kloeckner. Cutwidth and degeneracy of graphs. IF_PREPUB. 2009. HAL Id: hal-00408210 https://hal.archives-ouvertes.fr/hal-00408210v1
More informationPaths with two blocks in n-chromatic digraphs
Paths with two blocks in n-chromatic digraphs L. Addario-Berry, F. Havet and S. Thomassé September 20, 2005 Abstract We show that every oriented path of order n 4 with two blocks is contained in every
More informationThe Generalised Randić Index of Trees
The Generalised Randić Index of Trees Paul Balister Béla Bollobás Stefanie Gerke January 18, 2007 Abstract The Generalised Randić index R α (T) of a tree T is the sum over the edges uv of T of (d(u)d(v))
More informationOn the Turán number of forests
On the Turán number of forests Bernard Lidický Hong Liu Cory Palmer April 13, 01 Abstract The Turán number of a graph H, ex(n, H, is the maximum number of edges in a graph on n vertices which does not
More informationA simple branching process approach to the phase transition in G n,p
A simple branching process approach to the phase transition in G n,p Béla Bollobás Department of Pure Mathematics and Mathematical Statistics Wilberforce Road, Cambridge CB3 0WB, UK b.bollobas@dpmms.cam.ac.uk
More informationPacking nearly optimal Ramsey R(3, t) graphs
Packing nearly optimal Ramsey R(3, t) graphs He Guo Georgia Institute of Technology Joint work with Lutz Warnke Context of this talk Ramsey number R(s, t) R(s, t) := minimum n N such that every red/blue
More informationHOMEWORK #2 - MATH 3260
HOMEWORK # - MATH 36 ASSIGNED: JANUARAY 3, 3 DUE: FEBRUARY 1, AT :3PM 1) a) Give by listing the sequence of vertices 4 Hamiltonian cycles in K 9 no two of which have an edge in common. Solution: Here is
More informationOn Minimum Saturated Matrices
On Minimum Saturated Matrices Andrzej Dude Oleg Pihuro Andrew Thomason May 23, 2012 Abstract Motivated both by the wor of Anstee, Griggs, and Sali on forbidden submatrices and also by the extremal sat-function
More informationColouring powers of graphs with one cycle length forbidden
Colouring powers of graphs with one cycle length forbidden Ross J. Kang Radboud University Nijmegen CWI Networks & Optimization seminar 1/2017 Joint work with François Pirot. Example 1: ad hoc frequency
More informationPairs of Forbidden Subgraphs for Pancyclicity
1 / 24 Pairs of Forbidden Subgraphs for Pancyclicity James Carraher University of Nebraska Lincoln s-jcarrah1@math.unl.edu Joint Work with Mike Ferrara, Tim Morris, Mike Santana October 2012 Hamiltonian
More informationBipartite Subgraphs of Integer Weighted Graphs
Bipartite Subgraphs of Integer Weighted Graphs Noga Alon Eran Halperin February, 00 Abstract For every integer p > 0 let f(p be the minimum possible value of the maximum weight of a cut in an integer weighted
More informationForbidden Subgraphs for Pancyclicity
1 / 22 Forbidden Subgraphs for Pancyclicity James Carraher University of Nebraska Lincoln s-jcarrah1@math.unl.edu Joint Work with Mike Ferrara, Tim Morris, Mike Santana September 2012 Hamiltonian 2 / 22
More informationPRIME LABELING OF SMALL TREES WITH GAUSSIAN INTEGERS. 1. Introduction
PRIME LABELING OF SMALL TREES WITH GAUSSIAN INTEGERS HUNTER LEHMANN AND ANDREW PARK Abstract. A graph on n vertices is said to admit a prime labeling if we can label its vertices with the first n natural
More informationINDUCED CYCLES AND CHROMATIC NUMBER
INDUCED CYCLES AND CHROMATIC NUMBER A.D. SCOTT DEPARTMENT OF MATHEMATICS UNIVERSITY COLLEGE, GOWER STREET, LONDON WC1E 6BT Abstract. We prove that, for any pair of integers k, l 1, there exists an integer
More informationPartitioning a graph into highly connected subgraphs
Partitioning a graph into highly connected subgraphs Valentin Borozan 1,5, Michael Ferrara, Shinya Fujita 3 Michitaka Furuya 4, Yannis Manoussakis 5, Narayanan N 6 and Derrick Stolee 7 Abstract Given k
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 informationPartitioning 2-edge-colored Ore-type graphs by monochromatic cycles
Partitioning 2-edge-colored Ore-type graphs by monochromatic cycles János Barát MTA-ELTE Geometric and Algebraic Combinatorics Research Group barat@cs.elte.hu and Gábor N. Sárközy Alfréd Rényi Institute
More informationAn asymptotic multipartite Kühn-Osthus theorem
An asymptotic multipartite Kühn-Osthus theorem Ryan R. Martin 1 Richard Mycroft 2 Jozef Skokan 3 1 Iowa State University 2 University of Birmingham 3 London School of Economics 08 August 2017 Algebraic
More informationAALBORG UNIVERSITY. Total domination in partitioned graphs. Allan Frendrup, Preben Dahl Vestergaard and Anders Yeo
AALBORG UNIVERSITY Total domination in partitioned graphs by Allan Frendrup, Preben Dahl Vestergaard and Anders Yeo R-2007-08 February 2007 Department of Mathematical Sciences Aalborg University Fredrik
More informationPacking and decomposition of graphs with trees
Packing and decomposition of graphs with trees Raphael Yuster Department of Mathematics University of Haifa-ORANIM Tivon 36006, Israel. e-mail: raphy@math.tau.ac.il Abstract Let H be a tree on h 2 vertices.
More informationChromatic number, clique subdivisions, and the conjectures of Hajós and Erdős-Fajtlowicz
Chromatic number, clique subdivisions, and the conjectures of Hajós and Erdős-Fajtlowicz Jacob Fox Choongbum Lee Benny Sudakov Abstract For a graph G, let χ(g) denote its chromatic number and σ(g) denote
More informationRandom Graphs. Research Statement Daniel Poole Autumn 2015
I am interested in the interactions of probability and analysis with combinatorics and discrete structures. These interactions arise naturally when trying to find structure and information in large growing
More informationPaths with two blocks in n-chromatic digraphs
Paths with two blocks in n-chromatic digraphs Stéphan Thomassé, Frédéric Havet, Louigi Addario-Berry To cite this version: Stéphan Thomassé, Frédéric Havet, Louigi Addario-Berry. Paths with two blocks
More informationNote on Vertex-Disjoint Cycles
Note on Vertex-Disjoint Cycles Jacques Verstraëte Department of Pure Mathematics and Mathematical Statistics Centre for Mathematical Sciences Wilberforce Road, Cambridge CB3 OWB England. November 999.
More informationUpper Bounds of Dynamic Chromatic Number
Upper Bounds of Dynamic Chromatic Number Hong-Jian Lai, Bruce Montgomery and Hoifung Poon Department of Mathematics West Virginia University, Morgantown, WV 26506-6310 June 22, 2000 Abstract A proper vertex
More informationFast strategies in biased Maker Breaker games
Fast strategies in biased Maker Breaker games Mirjana Mikalački Miloš Stojaković July 11, 018 Abstract We study the biased (1 : b) Maker Breaker positional games, played on the edge set of the complete
More informationIntegrity in Graphs: Bounds and Basics
Integrity in Graphs: Bounds and Basics Wayne Goddard 1 Department of Mathematics Massachusetts Institute of Technology Cambridge, MA 02139 USA and Henda C. Swart Department of Mathematics, University of
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 informationA lower bound for the spectral radius of graphs with fixed diameter
A lower bound for the spectral radius of graphs with fixed diameter Sebastian M. Cioabă Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, USA e-mail: cioaba@math.udel.edu Edwin
More informationOn the bandwidth conjecture for 3-colourable graphs
On the bandwidth conjecture for 3-colourable graphs Julia Böttcher Technische Universität München Symposium on Discrete Algorithms, January 2007, New Orleans (joint work with Mathias Schacht & Anusch Taraz)
More informationBROOKS THEOREM AND BEYOND
BROOKS THEOREM AND BEYOND DANIEL W. CRANSTON AND LANDON RABERN Abstract. We collect some of our favorite proofs of Brooks Theorem, highlighting advantages and extensions of each. The proofs illustrate
More informationTurán numbers of expanded hypergraph forests
Turán numbers of expanded forests Rényi Institute of Mathematics, Budapest, Hungary Probabilistic and Extremal Combinatorics, IMA, Minneapolis, Sept. 9, 2014. Main results are joint with Tao JIANG, Miami
More informationPreliminaries. Graphs. E : set of edges (arcs) (Undirected) Graph : (i, j) = (j, i) (edges) V = {1, 2, 3, 4, 5}, E = {(1, 3), (3, 2), (2, 4)}
Preliminaries Graphs G = (V, E), V : set of vertices E : set of edges (arcs) (Undirected) Graph : (i, j) = (j, i) (edges) 1 2 3 5 4 V = {1, 2, 3, 4, 5}, E = {(1, 3), (3, 2), (2, 4)} 1 Directed Graph (Digraph)
More informationTiling on multipartite graphs
Tiling on multipartite graphs Ryan Martin Mathematics Department Iowa State University rymartin@iastate.edu SIAM Minisymposium on Graph Theory Joint Mathematics Meetings San Francisco, CA Ryan Martin (Iowa
More informationInduced subgraphs of prescribed size
Induced subgraphs of prescribed size Noga Alon Michael Krivelevich Benny Sudakov Abstract A subgraph of a graph G is called trivial if it is either a clique or an independent set. Let q(g denote the maximum
More informationMultiply Erdős-Ko-Rado Theorem
arxiv:1712.09942v1 [math.co] 28 Dec 2017 Multiply Erdős-Ko-Rado Theorem Carl Feghali Department of Informatics, University of Bergen, Bergen, Norway feghali.carl@gmail.com December 29, 2017 Abstract A
More informationJacques Verstraëte
2 - Turán s Theorem Jacques Verstraëte jacques@ucsd.edu 1 Introduction The aim of this section is to state and prove Turán s Theorem [17] and to discuss some of its generalizations, including the Erdős-Stone
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 informationLong cycles have the edge-erdős-pósa property
Long cycles have the edge-erdős-pósa property Henning Bruhn, Matthias Heinlein and Felix Joos Abstract We prove that the set of long cycles has the edge-erdős-pósa property: for every fixed integer l 3
More informationOn colorability of graphs with forbidden minors along paths and circuits
On colorability of graphs with forbidden minors along paths and circuits Elad Horev horevel@cs.bgu.ac.il Department of Computer Science Ben-Gurion University of the Negev Beer-Sheva 84105, Israel Abstract.
More informationThe 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?
More informationMinimum degree conditions for large subgraphs
Minimum degree conditions for large subgraphs Peter Allen 1 DIMAP University of Warwick Coventry, United Kingdom Julia Böttcher and Jan Hladký 2,3 Zentrum Mathematik Technische Universität München Garching
More informationEulerian Subgraphs in Graphs with Short Cycles
Eulerian Subgraphs in Graphs with Short Cycles Paul A. Catlin Hong-Jian Lai Abstract P. Paulraja recently showed that if every edge of a graph G lies in a cycle of length at most 5 and if G has no induced
More information10.4 The Kruskal Katona theorem
104 The Krusal Katona theorem 141 Example 1013 (Maximum weight traveling salesman problem We are given a complete directed graph with non-negative weights on edges, and we must find a maximum weight Hamiltonian
More informationEquitable coloring of random graphs
Michael Krivelevich 1, Balázs Patkós 2 1 Tel Aviv University, Tel Aviv, Israel 2 Central European University, Budapest, Hungary Phenomena in High Dimensions, Samos 2007 A set of vertices U V (G) is said
More informationOn a list-coloring problem
On a list-coloring problem Sylvain Gravier Frédéric Maffray Bojan Mohar December 24, 2002 Abstract We study the function f(g) defined for a graph G as the smallest integer k such that the join of G with
More informationRELATIVE N-TH NON-COMMUTING GRAPHS OF FINITE GROUPS. Communicated by Ali Reza Ashrafi. 1. Introduction
Bulletin of the Iranian Mathematical Society Vol. 39 No. 4 (2013), pp 663-674. RELATIVE N-TH NON-COMMUTING GRAPHS OF FINITE GROUPS A. ERFANIAN AND B. TOLUE Communicated by Ali Reza Ashrafi Abstract. Suppose
More informationAntoni Marczyk A NOTE ON ARBITRARILY VERTEX DECOMPOSABLE GRAPHS
Opuscula Mathematica Vol. 6 No. 1 006 Antoni Marczyk A NOTE ON ARBITRARILY VERTEX DECOMPOSABLE GRAPHS Abstract. A graph G of order n is said to be arbitrarily vertex decomposable if for each sequence (n
More informationCYCLES OF GIVEN SIZE IN A DENSE GRAPH
CYCLES OF GIVEN SIZE IN A DENSE GRAPH DANIEL J. HARVEY DAVID R. WOOD Abstract. We generalise a result of Corrádi and Hajnal and show that every graph with average degree at least 4 kr contains k vertex
More information