# Cycles and clique-minors in expanders

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Cycles and clique-minors in expanders Benny Sudakov UCLA and Princeton University

2 Expanders Definition: The vertex boundary of a subset X of a graph G: X = { all vertices in G\X with at least one neighbor in X }. Definition: Graph G on n vertices is an expander if X G of order at most n/2, X X is large (at least constant). G is locally expander if X X is large X G up to a certain size. Applications: Communication networks, Derandomization, Metric embeddings, Computational complexity, Coding theory, Markov chains,...

3 H-free graphs Definition: Graph G is called H-free if it contains no subgraph (not necessarily induced) isomorphic to H. Key observation: Let H be a fixed bipartite graph and let G be an H-free graph with minimum degree d. Then G is locally expanding. Examples of H: C 2k = cycle of length 2k. K s,t = complete bipartite graph with parts of size s t.

4 Expansion of H-free graphs Theorem: (Bondy-Simonovits, Kövári-Sós-Turán ) For every C 2k -free graph G on n vertices, e(g) c n 1+1/k. For every K s,t -free graph G on n vertices, e(g) c n 2 1/s. Corollary: If G is a C 2k -free graph with minimum degree d, then X > 2 X for all subsets X, X O(d k ). If G is a K s,t -free graph with minimum degree d, then X > 2 X for all subsets X, X O(d s/(s 1) ). Proof. If X 2 X and X is small, then Y = X X has size at most 3 X and at least d X /2 edges. This contradicts the theorem.

5 Cycle lengths Definition: cir(g) = circumference of graph G is the length of the longest cycle in G. C(G) = set of cycle lengths in G, i.e., all l such that C l G. Typical questions: How large is the circumference of G? How many different cycle lengths are in G? What are the arithmetic properties of C(G), e.g., is there a cycle in G whose length is a power of 2? Toy example: If G has minimum degree d then cir(g) d and C(G) d 1.

6 Graphs of large girth Definition: Girth of G is the length of the shortest cycle in G. Question: (Ore 1967) How large is cir(g) in graphs with min. degree d and girth 2k + 1? Known results: If G has minimum degree d and girth 2k + 1, then cir(g) Ω(kd) - Ore 1967 Improvements by Zhang, Zhao, Voss cir(g) Ω ( d k+1 2 ) - Ellingham and Menser 2000.

7 Many cycle lengths Conjecture: (Erdős 1992 ) If graph G has average degree d and girth 2k + 1, then C(G) Ω(d k ). Remarks: Tight for k = 2, 3, 5, and probably for all k. It is believed that for every fixed k there are graphs of order O(d k ) with minimum degree d and girth 2k + 1. True for k = 2 by Erdős, Faudree, Rousseau, and Schelp. Problem: (Erdős ) If G has large (but constant) average degree, does it contain a cycle of length 2 i for some i > 0?

8 New results Theorem: (S.- Verstraëte ) Let G be a graph with average degree d. If G is C 2k -free then C(G) Ω(d k ). ( If G is K s,t -free then C(G) Ω d s/(s 1) ). Moreover, C(G) contains an interval of consecutive even integers of this length. Theorem: (S.- Verstraëte ) If graph G on n vertices has average degree at least e c log n then G contains a cycle of length 2 i for some i > 0. (True for any exponentially growing sequence of even numbers)

9 Expansion and long path Lemma: (Pósa 1976 ) If every subset X G of size X m has X > 2 X then G contains a path of length 3m. Corollary: If G has average degree d and no cycle of length 2k then it contains a path of length Ω(d k ).

10 Existence of long cycle Let G be a connected, bipartite graph with average degree d and no cycle of length 2k. Fix any v G and let L i be all the vertices in G within distance i from v. Note that e(li, L i+1 ) = e(g) = d 2 G = d 2 Li d ( 4 Li + L i+1 ). Therefore one of the induced subgraphs G[L i, L i+1 ] has average degree at least d/2. It has a path of length Ω(d k ) between L i and L i+1, which together with v gives a long cycle.

11 Graph minors Definition: Graph Γ is a minor of graph G if for every vertex u Γ there is a connected subgraph G u G such that G u and G u are vertex disjoint for all u u. For every edge (u, u ) of Γ there is an edge from G u to G u.

12 Clique minors Question: Find sufficient condition for graph G to contain given Γ as a minor. (e.g., when Γ is a clique) Conjecture: (Hadwiger 1943) Every graph with chromatic number at least k contains a clique on k vertices as a minor. Remark: Every graph with chromatic number k contains a subgraph with minimum degree k 1. Theorem: (Kostochka, Thomason 80 s) Every graph with average degree d contains a clique-minor of order d c log d.

13 Minors in graphs of large girth Theorem: (Thomassen, Diestel-Rompel, Kühn-Osthus) If G has girth 2k + 1 and average degree d, then it contains a minor with average degree Ω ( ) d k+1 2. Observation: Any minor of graph G has at most e(g) edges and therefore has average degree at most 2e(G). Remark: Graphs of order O(d k ), with minimum degree d and girth 2k + 1, have O(d k+1 ) edges, so they cannot have minors with average degree d k+1 2. Question: Do we need the girth assumption? Is it enough to forbid cycle C 2k?

14 Minors in H-free graphs Theorem: (Kühn-Osthus) Every K s,t -free graph G with average degree d contains a minor ) with average degree Ω. ( 1+ 1 d 2(s 1) polylog d Remarks: This implies that H-free graphs (for bipartite H) satisfy Hadwiger s conjecture. There is a K s,t -free graph on O ( d s/(s 1)) vertices, with average degree d. It has O ( d 2+1/(s 1)) edges and cannot contain a minor with average degree d (s 1). Conjecture: (Kühn-Osthus) Every K s,t -free graph with average degree d contains a minor with average degree Ω ( d 1+ 1 ) 2(s 1).

15 New results Theorem: (Krivelevich-S.) Let G be a graph with average degree d. If G is C 2k -free then it contains a minor with average degree Ω ( d k+1 2 If G is K s,t -free then it contains a minor with average degree ). Ω ( d 1+ 2(s 1)) 1. Remark: The same approach is used to prove both statements. One of its key ingredients is the expansion property of H-free graphs.

16 Minors in expanders Definition: A separator of a graph G of order n is a set of vertices whose removal separates G into connected components of size 2n/3. Theorem: (Plotkin-Rao-Smith, improving on Alon-Seymour-Thomas) If an n-vertex graph G has no clique-minor on h vertices, then it has a separator of order O ( h n log n ). Corollary: (Plotkin-Rao-Smith, Kleinberg-Rubinfeld) Let G be a graph on n vertices and let t > 0 be a constant. If all subsets X G of size at most n/2( have X ) t X then G contains a clique-minor of order Ω n log n.

17 Superconstant expansion Question: What size clique-minors can one find in a graph G on n vertices, whose expansion factor t together with n? Remarks: Proof based on separators can t give bound better than n log n since every graph of order n has separator of size n/3. Random graph G(n, t/n) has almost surely expansion factor t and clique-minor of order Ω ( nt log n). (Bollobás-Erdős-Catlin, improved by Fountoulakis-Kühn-Osthus) Question: What if edges of G are distributed like in the random graph?

18 Expansion of C 4 -free graphs revisited Proposition: Let G be a C 4 -free graph with minimum degree d, and let X be a subset of G of size X d. Then X d 3 X. Proof. Suppose X < d 3 X. Since e(x, X ) = (1 + o(1))d X, y X ( ) dx (y) X 2 ( e(x, X ) X 2 ) ( X d X 2 ). This gives a 4-cycle, contradiction. Remarks: If G is a C 4 -free graph with n = O(d 2 ) vertices and minimum degree d, then it has expansion factor t = d 3 = Ω( n). There are similar results for all H-free graphs (H bipartite).

19 New result: expanders Theorem: (Krivelevich-S.) If all subsets X G of size X O(n/t) have X t X then G contains a minor with average degree ( ) Ω nt log t log n. Remarks: For t = n ɛ this gives a minor with average degree Ω( nt) and thus also a clique-minor of order Ω( nt/ log n). The random graph G(n, t/n) shows that this bound is tight. If G is a C 4 -free graph of order n = O(d 2 ), then for t = d/3 every subset of G of size O(n/t) expands by factor of t. Thus G has a minor with average degree Ω( nt) = Ω(d 3/2 )

20 New result: pseudo-random graphs Definition: A graph G is (p, β)-jumbled if for every subset of vertices X e(x ) p X 2 /2 β X. Theorem: (Krivelevich-S.) Every (p, β)-jumbled graph G on n vertices with β = o(np) contains a minor with average degree Ω(n p). Remarks: This is tight since such a graph G has O(n 2 p) edges. It extends results of Thomason and Drier-Linial. Expansion factor of (p, β)-jumbled G can be only np β np. If G is an (n, d, λ)-graph then it is (d/n, λ)-jumbled. Hence if λ = o(d) it has a minor with average degree Ω( nd).

### A Turán Type Problem Concerning the Powers of the Degrees of a Graph

A Turán Type Problem Concerning the Powers of the Degrees of a Graph Yair Caro and Raphael Yuster Department of Mathematics University of Haifa-ORANIM, Tivon 36006, Israel. AMS Subject Classification:

### Graphs without proper subgraphs of minimum degree 3 and short cycles

Graphs without proper subgraphs of minimum degree 3 and short cycles Lothar Narins, Alexey Pokrovskiy, Tibor Szabó Department of Mathematics, Freie Universität, Berlin, Germany. August 22, 2014 Abstract

### Minimum degree condition forcing complete graph immersion

Minimum degree condition forcing complete graph immersion Matt DeVos Department of Mathematics Simon Fraser University Burnaby, B.C. V5A 1S6 Jacob Fox Department of Mathematics MIT Cambridge, MA 02139

### Embedding nearly-spanning bounded degree trees

Embedding nearly-spanning bounded degree trees Noga Alon Michael Krivelevich Benny Sudakov Abstract We derive a sufficient condition for a sparse graph G on n vertices to contain a copy of a tree T of

### A 2-factor in which each cycle has long length in claw-free graphs

A -factor in which each cycle has long length in claw-free graphs Roman Čada Shuya Chiba Kiyoshi Yoshimoto 3 Department of Mathematics University of West Bohemia and Institute of Theoretical Computer Science

### Labeling outerplanar graphs with maximum degree three

Labeling outerplanar graphs with maximum degree three Xiangwen Li 1 and Sanming Zhou 2 1 Department of Mathematics Huazhong Normal University, Wuhan 430079, China 2 Department of Mathematics and Statistics

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

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

### Every tree contains a large induced subgraph with all degrees odd

Every tree contains a large induced subgraph with all degrees odd A.J. Radcliffe Carnegie Mellon University, Pittsburgh, PA A.D. Scott Department of Pure Mathematics and Mathematical Statistics University

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

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

### Induced subgraphs of Ramsey graphs with many distinct degrees

Journal of Combinatorial Theory, Series B 97 (2007) 62 69 www.elsevier.com/locate/jctb Induced subgraphs of Ramsey graphs with many distinct degrees Boris Bukh a, Benny Sudakov a,b a Department of Mathematics,

### On the number-theoretic functions ν(n) and Ω(n)

ACTA ARITHMETICA LXXVIII.1 (1996) On the number-theoretic functions ν(n) and Ω(n) by Jiahai Kan (Nanjing) 1. Introduction. Let d(n) denote the divisor function, ν(n) the number of distinct prime factors,

### Mean Ramsey-Turán numbers

Mean Ramsey-Turán numbers Raphael Yuster Department of Mathematics University of Haifa at Oranim Tivon 36006, Israel Abstract A ρ-mean coloring of a graph is a coloring of the edges such that the average

### Ph.D. Thesis. Judit Nagy-György. Supervisor: Péter Hajnal Associate Professor

Online algorithms for combinatorial problems Ph.D. Thesis by Judit Nagy-György Supervisor: Péter Hajnal Associate Professor Doctoral School in Mathematics and Computer Science University of Szeged Bolyai

### Total colorings of planar graphs with small maximum degree

Total colorings of planar graphs with small maximum degree Bing Wang 1,, Jian-Liang Wu, Si-Feng Tian 1 Department of Mathematics, Zaozhuang University, Shandong, 77160, China School of Mathematics, Shandong

### Large induced subgraphs with all degrees odd

Large induced subgraphs with all degrees odd A.D. Scott Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, England Abstract: We prove that every connected graph of order

### Extremal Graph Theory

Extremal Graph Theory Instructor: Asaf Shapira Scribed by Guy Rutenberg Fall 013 Contents 1 First Lecture 1.1 Mantel s Theorem........................................ 3 1. Turán s Theorem.........................................

### ON INDUCED SUBGRAPHS WITH ALL DEGREES ODD. 1. Introduction

ON INDUCED SUBGRAPHS WITH ALL DEGREES ODD A.D. SCOTT Abstract. Gallai proved that the vertex set of any graph can be partitioned into two sets, each inducing a subgraph with all degrees even. We prove

### SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH

31 Kragujevac J. Math. 25 (2003) 31 49. SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH Kinkar Ch. Das Department of Mathematics, Indian Institute of Technology, Kharagpur 721302, W.B.,

### Divisor graphs have arbitrary order and size

Divisor graphs have arbitrary order and size arxiv:math/0606483v1 [math.co] 20 Jun 2006 Le Anh Vinh School of Mathematics University of New South Wales Sydney 2052 Australia Abstract A divisor graph G

### The degree, size and chromatic index of a uniform hypergraph

The degree, size and chromatic index of a uniform hypergraph Noga Alon Jeong Han Kim Abstract Let H be a k-uniform hypergraph in which no two edges share more than t common vertices, and let D denote the

### Introduced by Stuart Kauffman (ca. 1986) as a tunable family of fitness landscapes.

68 Part II. Combinatorial Models can require a number of spin flips that is exponential in N (A. Haken et al. ca. 1989), and that one can in fact embed arbitrary computations in the dynamics (Orponen 1995).

### GRAPH THEORY LECTURE 4: TREES

GRAPH THEORY LECTURE 4: TREES Abstract. 3.1 presents some standard characterizations and properties of trees. 3.2 presents several different types of trees. 3.7 develops a counting method based on a bijection

### A Sublinear Bipartiteness Tester for Bounded Degree Graphs

A Sublinear Bipartiteness Tester for Bounded Degree Graphs Oded Goldreich Dana Ron February 5, 1998 Abstract We present a sublinear-time algorithm for testing whether a bounded degree graph is bipartite

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

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

### Maximum Hitting Time for Random Walks on Graphs. Graham Brightwell, Cambridge University Peter Winkler*, Emory University

Maximum Hitting Time for Random Walks on Graphs Graham Brightwell, Cambridge University Peter Winkler*, Emory University Abstract. For x and y vertices of a connected graph G, let T G (x, y denote the

### Cycle transversals in bounded degree graphs

Electronic Notes in Discrete Mathematics 35 (2009) 189 195 www.elsevier.com/locate/endm Cycle transversals in bounded degree graphs M. Groshaus a,2,3 P. Hell b,3 S. Klein c,1,3 L. T. Nogueira d,1,3 F.

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

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

### RAMSEY-TYPE NUMBERS FOR DEGREE SEQUENCES

RAMSEY-TYPE NUMBERS FOR DEGREE SEQUENCES ARTHUR BUSCH UNIVERSITY OF DAYTON DAYTON, OH 45469 MICHAEL FERRARA, MICHAEL JACOBSON 1 UNIVERSITY OF COLORADO DENVER DENVER, CO 80217 STEPHEN G. HARTKE 2 UNIVERSITY

### An inequality for the group chromatic number of a graph

An inequality for the group chromatic number of a graph Hong-Jian Lai 1, Xiangwen Li 2 and Gexin Yu 3 1 Department of Mathematics, West Virginia University Morgantown, WV 26505 USA 2 Department of Mathematics

### CS311H. Prof: Peter Stone. Department of Computer Science The University of Texas at Austin

CS311H Prof: Department of Computer Science The University of Texas at Austin Good Morning, Colleagues Good Morning, Colleagues Are there any questions? Logistics Class survey Logistics Class survey Homework

### Ramsey numbers for bipartite graphs with small bandwidth

Ramsey numbers for bipartite graphs with small bandwidth Guilherme O. Mota 1,, Gábor N. Sárközy 2,, Mathias Schacht 3,, and Anusch Taraz 4, 1 Instituto de Matemática e Estatística, Universidade de São

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

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

### Exponential time algorithms for graph coloring

Exponential time algorithms for graph coloring Uriel Feige Lecture notes, March 14, 2011 1 Introduction Let [n] denote the set {1,..., k}. A k-labeling of vertices of a graph G(V, E) is a function V [k].

### Complexity Theory. IE 661: Scheduling Theory Fall 2003 Satyaki Ghosh Dastidar

Complexity Theory IE 661: Scheduling Theory Fall 2003 Satyaki Ghosh Dastidar Outline Goals Computation of Problems Concepts and Definitions Complexity Classes and Problems Polynomial Time Reductions Examples

### M-Degrees of Quadrangle-Free Planar Graphs

M-Degrees of Quadrangle-Free Planar Graphs Oleg V. Borodin, 1 Alexandr V. Kostochka, 1,2 Naeem N. Sheikh, 2 and Gexin Yu 3 1 SOBOLEV INSTITUTE OF MATHEMATICS NOVOSIBIRSK 630090, RUSSIA E-mail: brdnoleg@math.nsc.ru

### Definition 11.1. Given a graph G on n vertices, we define the following quantities:

Lecture 11 The Lovász ϑ Function 11.1 Perfect graphs We begin with some background on perfect graphs. graphs. First, we define some quantities on Definition 11.1. Given a graph G on n vertices, we define

### A threshold for the Maker-Breaker clique game

A threshold for the Maker-Breaker clique game Tobias Müller Miloš Stojaković October 7, 01 Abstract We study the Maker-Breaker k-clique game played on the edge set of the random graph G(n, p. In this game,

### All trees contain a large induced subgraph having all degrees 1 (mod k)

All trees contain a large induced subgraph having all degrees 1 (mod k) David M. Berman, A.J. Radcliffe, A.D. Scott, Hong Wang, and Larry Wargo *Department of Mathematics University of New Orleans New

### Nimble Algorithms for Cloud Computing. Ravi Kannan, Santosh Vempala and David Woodruff

Nimble Algorithms for Cloud Computing Ravi Kannan, Santosh Vempala and David Woodruff Cloud computing Data is distributed arbitrarily on many servers Parallel algorithms: time Streaming algorithms: sublinear

### Random graphs with a given degree sequence

Sourav Chatterjee (NYU) Persi Diaconis (Stanford) Allan Sly (Microsoft) Let G be an undirected simple graph on n vertices. Let d 1,..., d n be the degrees of the vertices of G arranged in descending order.

### The positive minimum degree game on sparse graphs

The positive minimum degree game on sparse graphs József Balogh Department of Mathematical Sciences University of Illinois, USA jobal@math.uiuc.edu András Pluhár Department of Computer Science University

### High degree graphs contain large-star factors

High degree graphs contain large-star factors Dedicated to László Lovász, for his 60th birthday Noga Alon Nicholas Wormald Abstract We show that any finite simple graph with minimum degree d contains a

### 10. Graph Matrices Incidence Matrix

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

### Small Maximal Independent Sets and Faster Exact Graph Coloring

Small Maximal Independent Sets and Faster Exact Graph Coloring David Eppstein Univ. of California, Irvine Dept. of Information and Computer Science The Exact Graph Coloring Problem: Given an undirected

### Stationary random graphs on Z with prescribed iid degrees and finite mean connections

Stationary random graphs on Z with prescribed iid degrees and finite mean connections Maria Deijfen Johan Jonasson February 2006 Abstract Let F be a probability distribution with support on the non-negative

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

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

### 1 Definitions. Supplementary Material for: Digraphs. Concept graphs

Supplementary Material for: van Rooij, I., Evans, P., Müller, M., Gedge, J. & Wareham, T. (2008). Identifying Sources of Intractability in Cognitive Models: An Illustration using Analogical Structure Mapping.

### 2.3 Scheduling jobs on identical parallel machines

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

### Regular Induced Subgraphs of a Random Graph*

Regular Induced Subgraphs of a Random Graph* Michael Krivelevich, 1 Benny Sudaov, Nicholas Wormald 3 1 School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel; e-mail: rivelev@post.tau.ac.il

### The chromatic spectrum of mixed hypergraphs

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

### The Goldberg Rao Algorithm for the Maximum Flow Problem

The Goldberg Rao Algorithm for the Maximum Flow Problem COS 528 class notes October 18, 2006 Scribe: Dávid Papp Main idea: use of the blocking flow paradigm to achieve essentially O(min{m 2/3, n 1/2 }

### Best Monotone Degree Bounds for Various Graph Parameters

Best Monotone Degree Bounds for Various Graph Parameters D. Bauer Department of Mathematical Sciences Stevens Institute of Technology Hoboken, NJ 07030 S. L. Hakimi Department of Electrical and Computer

### Single machine parallel batch scheduling with unbounded capacity

Workshop on Combinatorics and Graph Theory 21th, April, 2006 Nankai University Single machine parallel batch scheduling with unbounded capacity Yuan Jinjiang Department of mathematics, Zhengzhou University

### On-line Ramsey numbers

On-line Ramsey numbers David Conlon Abstract Consider the following game between two players, Builder and Painter Builder draws edges one at a time and Painter colours them, in either red or blue, as each

### Discrete Mathematics & Mathematical Reasoning Chapter 10: Graphs

Discrete Mathematics & Mathematical Reasoning Chapter 10: Graphs Kousha Etessami U. of Edinburgh, UK Kousha Etessami (U. of Edinburgh, UK) Discrete Mathematics (Chapter 6) 1 / 13 Overview Graphs and Graph

### Generalized Induced Factor Problems

Egerváry Research Group on Combinatorial Optimization Technical reports TR-2002-07. Published by the Egrerváry Research Group, Pázmány P. sétány 1/C, H 1117, Budapest, Hungary. Web site: www.cs.elte.hu/egres.

### Fundamentals of Media Theory

Fundamentals of Media Theory ergei Ovchinnikov Mathematics Department an Francisco tate University an Francisco, CA 94132 sergei@sfsu.edu Abstract Media theory is a new branch of discrete applied mathematics

### Testing Hereditary Properties of Non-Expanding Bounded-Degree Graphs

Testing Hereditary Properties of Non-Expanding Bounded-Degree Graphs Artur Czumaj Asaf Shapira Christian Sohler Abstract We study graph properties which are testable for bounded degree graphs in time independent

### On an anti-ramsey type result

On an anti-ramsey type result Noga Alon, Hanno Lefmann and Vojtĕch Rödl Abstract We consider anti-ramsey type results. For a given coloring of the k-element subsets of an n-element set X, where two k-element

### Discrete Mathematics Problems

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

### 3. Eulerian and Hamiltonian Graphs

3. Eulerian and Hamiltonian Graphs There are many games and puzzles which can be analysed by graph theoretic concepts. In fact, the two early discoveries which led to the existence of graphs arose from

### most 4 Mirka Miller 1,2, Guillermo Pineda-Villavicencio 3, The University of Newcastle Callaghan, NSW 2308, Australia University of West Bohemia

Complete catalogue of graphs of maimum degree 3 and defect at most 4 Mirka Miller 1,2, Guillermo Pineda-Villavicencio 3, 1 School of Electrical Engineering and Computer Science The University of Newcastle

### On the independence number of graphs with maximum degree 3

On the independence number of graphs with maximum degree 3 Iyad A. Kanj Fenghui Zhang Abstract Let G be an undirected graph with maximum degree at most 3 such that G does not contain any of the three graphs

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

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

### Pseudo-random Graphs. 1. Introduction. M. KRIVELEVICH and B. SUDAKOV

BOLYAI SOCIETY MATHEMATICAL STUDIES, X Conference on Finite and Infinite Sets Budapest, pp. 1 64. Pseudo-random Graphs M. KRIVELEVICH and B. SUDAKOV 1. Introduction Random graphs have proven to be one

### Cacti with minimum, second-minimum, and third-minimum Kirchhoff indices

MATHEMATICAL COMMUNICATIONS 47 Math. Commun., Vol. 15, No. 2, pp. 47-58 (2010) Cacti with minimum, second-minimum, and third-minimum Kirchhoff indices Hongzhuan Wang 1, Hongbo Hua 1, and Dongdong Wang

### arxiv:1112.0829v1 [math.pr] 5 Dec 2011

How Not to Win a Million Dollars: A Counterexample to a Conjecture of L. Breiman Thomas P. Hayes arxiv:1112.0829v1 [math.pr] 5 Dec 2011 Abstract Consider a gambling game in which we are allowed to repeatedly

### Lecture 15 An Arithmetic Circuit Lowerbound and Flows in Graphs

CSE599s: Extremal Combinatorics November 21, 2011 Lecture 15 An Arithmetic Circuit Lowerbound and Flows in Graphs Lecturer: Anup Rao 1 An Arithmetic Circuit Lower Bound An arithmetic circuit is just like

### On end degrees and infinite cycles in locally finite graphs

On end degrees and infinite cycles in locally finite graphs Henning Bruhn Maya Stein Abstract We introduce a natural extension of the vertex degree to ends. For the cycle space C(G) as proposed by Diestel

### Analysis of Algorithms, I

Analysis of Algorithms, I CSOR W4231.002 Eleni Drinea Computer Science Department Columbia University Thursday, February 26, 2015 Outline 1 Recap 2 Representing graphs 3 Breadth-first search (BFS) 4 Applications

### The average distances in random graphs with given expected degrees

Classification: Physical Sciences, Mathematics The average distances in random graphs with given expected degrees by Fan Chung 1 and Linyuan Lu Department of Mathematics University of California at San

### 4. Expanding dynamical systems

4.1. Metric definition. 4. Expanding dynamical systems Definition 4.1. Let X be a compact metric space. A map f : X X is said to be expanding if there exist ɛ > 0 and L > 1 such that d(f(x), f(y)) Ld(x,

### Approximation Algorithms: LP Relaxation, Rounding, and Randomized Rounding Techniques. My T. Thai

Approximation Algorithms: LP Relaxation, Rounding, and Randomized Rounding Techniques My T. Thai 1 Overview An overview of LP relaxation and rounding method is as follows: 1. Formulate an optimization

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

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

### SEMITOTAL AND TOTAL BLOCK-CUTVERTEX GRAPH

CHAPTER 3 SEMITOTAL AND TOTAL BLOCK-CUTVERTEX GRAPH ABSTRACT This chapter begins with the notion of block distances in graphs. Using block distance we defined the central tendencies of a block, like B-radius

### Removing Local Extrema from Imprecise Terrains

Removing Local Extrema from Imprecise Terrains Chris Gray Frank Kammer Maarten Löffler Rodrigo I. Silveira Abstract In this paper we consider imprecise terrains, that is, triangulated terrains with a vertical

### CSC2420 Fall 2012: Algorithm Design, Analysis and Theory

CSC2420 Fall 2012: Algorithm Design, Analysis and Theory Allan Borodin November 15, 2012; Lecture 10 1 / 27 Randomized online bipartite matching and the adwords problem. We briefly return to online algorithms

### Computer Algorithms. NP-Complete Problems. CISC 4080 Yanjun Li

Computer Algorithms NP-Complete Problems NP-completeness The quest for efficient algorithms is about finding clever ways to bypass the process of exhaustive search, using clues from the input in order

### 8. Matchings and Factors

8. Matchings and Factors Consider the formation of an executive council by the parliament committee. Each committee needs to designate one of its members as an official representative to sit on the council,

### P versus NP, and More

1 P versus NP, and More Great Ideas in Theoretical Computer Science Saarland University, Summer 2014 If you have tried to solve a crossword puzzle, you know that it is much harder to solve it than to verify

### An inequality for the group chromatic number of a graph

Discrete Mathematics 307 (2007) 3076 3080 www.elsevier.com/locate/disc Note An inequality for the group chromatic number of a graph Hong-Jian Lai a, Xiangwen Li b,,1, Gexin Yu c a Department of Mathematics,

### On the crossing number of K m,n

On the crossing number of K m,n Nagi H. Nahas nnahas@acm.org Submitted: Mar 15, 001; Accepted: Aug 10, 00; Published: Aug 1, 00 MR Subject Classifications: 05C10, 05C5 Abstract The best lower bound known

### Approximated Distributed Minimum Vertex Cover Algorithms for Bounded Degree Graphs

Approximated Distributed Minimum Vertex Cover Algorithms for Bounded Degree Graphs Yong Zhang 1.2, Francis Y.L. Chin 2, and Hing-Fung Ting 2 1 College of Mathematics and Computer Science, Hebei University,

### On three zero-sum Ramsey-type problems

On three zero-sum Ramsey-type problems Noga Alon Department of Mathematics Raymond and Beverly Sackler Faculty of Exact Sciences Tel Aviv University, Tel Aviv, Israel and Yair Caro Department of Mathematics

### A DIRAC TYPE RESULT ON HAMILTON CYCLES IN ORIENTED GRAPHS

A DIRAC TYPE RESULT ON HAMILTON CYCLES IN ORIENTED GRAPHS LUKE KELLY, DANIELA KÜHN AND DERYK OSTHUS Abstract. We show that for each α > every sufficiently large oriented graph G with δ + (G), δ (G) 3 G

### ZERO-DIVISOR GRAPHS OF POLYNOMIALS AND POWER SERIES OVER COMMUTATIVE RINGS

ZERO-DIVISOR GRAPHS OF POLYNOMIALS AND POWER SERIES OVER COMMUTATIVE RINGS M. AXTELL, J. COYKENDALL, AND J. STICKLES Abstract. We recall several results of zero divisor graphs of commutative rings. We

### 1. Nondeterministically guess a solution (called a certificate) 2. Check whether the solution solves the problem (called verification)

Some N P problems Computer scientists have studied many N P problems, that is, problems that can be solved nondeterministically in polynomial time. Traditionally complexity question are studied as languages:

### On Some Vertex Degree Based Graph Invariants

MATCH Communications in Mathematical and in Computer Chemistry MATCH Commun. Math. Comput. Chem. 65 (20) 723-730 ISSN 0340-6253 On Some Vertex Degree Based Graph Invariants Batmend Horoldagva a and Ivan

### Balloons, Cut-Edges, Matchings, and Total Domination in Regular Graphs of Odd Degree

Balloons, Cut-Edges, Matchings, and Total Domination in Regular Graphs of Odd Degree Suil O, Douglas B. West November 9, 2008; revised June 2, 2009 Abstract A balloon in a graph G is a maximal 2-edge-connected

### 1 Introduction. Dr. T. Srinivas Department of Mathematics Kakatiya University Warangal 506009, AP, INDIA tsrinivasku@gmail.com

A New Allgoriitthm for Miiniimum Costt Liinkiing M. Sreenivas Alluri Institute of Management Sciences Hanamkonda 506001, AP, INDIA allurimaster@gmail.com Dr. T. Srinivas Department of Mathematics Kakatiya

### Markov random fields and Gibbs measures

Chapter Markov random fields and Gibbs measures 1. Conditional independence Suppose X i is a random element of (X i, B i ), for i = 1, 2, 3, with all X i defined on the same probability space (.F, P).

### Graphical degree sequences and realizations

swap Graphical and realizations Péter L. Erdös Alfréd Rényi Institute of Mathematics Hungarian Academy of Sciences MAPCON 12 MPIPKS - Dresden, May 15, 2012 swap Graphical and realizations Péter L. Erdös

### Introduction to Graph Theory

Introduction to Graph Theory Allen Dickson October 2006 1 The Königsberg Bridge Problem The city of Königsberg was located on the Pregel river in Prussia. The river divided the city into four separate

### 5 Directed acyclic graphs

5 Directed acyclic graphs (5.1) Introduction In many statistical studies we have prior knowledge about a temporal or causal ordering of the variables. In this chapter we will use directed graphs to incorporate

### UPPER BOUNDS ON THE L(2, 1)-LABELING NUMBER OF GRAPHS WITH MAXIMUM DEGREE

UPPER BOUNDS ON THE L(2, 1)-LABELING NUMBER OF GRAPHS WITH MAXIMUM DEGREE ANDREW LUM ADVISOR: DAVID GUICHARD ABSTRACT. L(2,1)-labeling was first defined by Jerrold Griggs [Gr, 1992] as a way to use graphs

### 1. Prove that the empty set is a subset of every set.

1. Prove that the empty set is a subset of every set. Basic Topology Written by Men-Gen Tsai email: b89902089@ntu.edu.tw Proof: For any element x of the empty set, x is also an element of every set since

### arxiv: v2 [math.co] 30 Nov 2015

PLANAR GRAPH IS ON FIRE PRZEMYSŁAW GORDINOWICZ arxiv:1311.1158v [math.co] 30 Nov 015 Abstract. Let G be any connected graph on n vertices, n. Let k be any positive integer. Suppose that a fire breaks out

### Degree Hypergroupoids Associated with Hypergraphs

Filomat 8:1 (014), 119 19 DOI 10.98/FIL1401119F Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Degree Hypergroupoids Associated

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

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

### Game Chromatic Index of Graphs with Given Restrictions on Degrees

Game Chromatic Index of Graphs with Given Restrictions on Degrees Andrew Beveridge Department of Mathematics and Computer Science Macalester College St. Paul, MN 55105 Tom Bohman, Alan Frieze, and Oleg