A Guide to the Discharging Method
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On the Tree–Depth of Random Graphs Arxiv:1104.2132V2 [Math.CO] 15 Feb 2012
On the tree–depth of random graphs ∗ G. Perarnau and O.Serra November 11, 2018 Abstract The tree–depth is a parameter introduced under several names as a measure of sparsity of a graph. We compute asymptotic values of the tree–depth of random graphs. For dense graphs, p n−1, the tree–depth of a random graph G is a.a.s. td(G) = n − O(pn=p). Random graphs with p = c=n, have a.a.s. linear tree–depth when c > 1, the tree–depth is Θ(log n) when c = 1 and Θ(log log n) for c < 1. The result for c > 1 is derived from the computation of tree–width and provides a more direct proof of a conjecture by Gao on the linearity of tree–width recently proved by Lee, Lee and Oum [?]. We also show that, for c = 1, every width parameter is a.a.s. constant, and that random regular graphs have linear tree–depth. 1 Introduction An elimination tree of a graph G is a rooted tree on the set of vertices such that there are no edges in G between vertices in different branches of the tree. The natural elimination scheme provided by this tree is used in many graph algorithmic problems where two non adjacent subsets of vertices can be managed independently. One good example is the Cholesky decomposition of symmetric matrices (see [?, ?, ?, ?]). Given an elimination tree, a distributed algorithm can be designed which takes care of disjoint subsets of vertices in different parallel processors. Starting arXiv:1104.2132v2 [math.CO] 15 Feb 2012 by the furthest level from the root, it proceeds by exposing at each step the vertices at a given depth. -
Testing Isomorphism in the Bounded-Degree Graph Model (Preliminary Version)
Electronic Colloquium on Computational Complexity, Revision 1 of Report No. 102 (2019) Testing Isomorphism in the Bounded-Degree Graph Model (preliminary version) Oded Goldreich∗ August 11, 2019 Abstract We consider two versions of the problem of testing graph isomorphism in the bounded-degree graph model: A version in which one graph is fixed, and a version in which the input consists of two graphs. We essentially determine the query complexity of these testing problems in the special case of n-vertex graphs with connected components of size at most poly(log n). This is done by showing that these problems are computationally equivalent (up to polylogarithmic factors) to corresponding problems regarding isomorphism between sequences (over a large alphabet). Ignoring the dependence on the proximity parameter, our main results are: 1. The query complexity of testing isomorphism to a fixed object (i.e., an n-vertex graph or an n-long sequence) is Θ(e n1=2). 2. The query complexity of testing isomorphism between two input objects is Θ(e n2=3). Testing isomorphism between two sequences is shown to be related to testing that two distributions are equivalent, and this relation yields reductions in three of the four relevant cases. Failing to reduce the problem of testing the equivalence of two distribution to the problem of testing isomorphism between two sequences, we adapt the proof of the lower bound on the complexity of the first problem to the second problem. This adaptation constitutes the main technical contribution of the current work. Determining the complexity of testing graph isomorphism (in the bounded-degree graph model), in the general case (i.e., for arbitrary bounded-degree graphs), is left open. -
Practical Verification of MSO Properties of Graphs of Bounded Clique-Width
Practical verification of MSO properties of graphs of bounded clique-width Ir`ene Durand (joint work with Bruno Courcelle) LaBRI, Universit´ede Bordeaux 20 Octobre 2010 D´ecompositions de graphes, th´eorie, algorithmes et logiques, 2010 Objectives Verify properties of graphs of bounded clique-width Properties ◮ connectedness, ◮ k-colorability, ◮ existence of cycles ◮ existence of paths ◮ bounds (cardinality, degree, . ) ◮ . How : using term automata Note that we consider finite graphs only 2/33 Graphs as relational structures For simplicity, we consider simple,loop-free undirected graphs Extensions are easy Every graph G can be identified with the relational structure (VG , edgG ) where VG is the set of vertices and edgG ⊆VG ×VG the binary symmetric relation that defines edges. v7 v6 v2 v8 v1 v3 v5 v4 VG = {v1, v2, v3, v4, v5, v6, v7, v8} edgG = {(v1, v2), (v1, v4), (v1, v5), (v1, v7), (v2, v3), (v2, v6), (v3, v4), (v4, v5), (v5, v8), (v6, v7), (v7, v8)} 3/33 Expression of graph properties First order logic (FO) : ◮ quantification on single vertices x, y . only ◮ too weak ; can only express ”local” properties ◮ k-colorability (k > 1) cannot be expressed Second order logic (SO) ◮ quantifications on relations of arbitrary arity ◮ SO can express most properties of interest in Graph Theory ◮ too complex (many problems are undecidable or do not have a polynomial solution). 4/33 Monadic second order logic (MSO) ◮ SO formulas that only use quantifications on unary relations (i.e., on sets). ◮ can express many useful graph properties like connectedness, k-colorability, planarity... Example : k-colorability Stable(X ) : ∀u, v(u ∈ X ∧ v ∈ X ⇒¬edg(u, v)) Partition(X1,..., Xm) : ∀x(x ∈ X1 ∨ . -
Harary Polynomials
numerative ombinatorics A pplications Enumerative Combinatorics and Applications ECA 1:2 (2021) Article #S2R13 ecajournal.haifa.ac.il Harary Polynomials Orli Herscovici∗, Johann A. Makowskyz and Vsevolod Rakitay ∗Department of Mathematics, University of Haifa, Israel Email: [email protected] zDepartment of Computer Science, Technion{Israel Institute of Technology, Haifa, Israel Email: [email protected] yDepartment of Mathematics, Technion{Israel Institute of Technology, Haifa, Israel Email: [email protected] Received: November 13, 2020, Accepted: February 7, 2021, Published: February 19, 2021 The authors: Released under the CC BY-ND license (International 4.0) Abstract: Given a graph property P, F. Harary introduced in 1985 P-colorings, graph colorings where each color class induces a graph in P. Let χP (G; k) counts the number of P-colorings of G with at most k colors. It turns out that χP (G; k) is a polynomial in Z[k] for each graph G. Graph polynomials of this form are called Harary polynomials. In this paper we investigate properties of Harary polynomials and compare them with properties of the classical chromatic polynomial χ(G; k). We show that the characteristic and the Laplacian polynomial, the matching, the independence and the domination polynomials are not Harary polynomials. We show that for various notions of sparse, non-trivial properties P, the polynomial χP (G; k) is, in contrast to χ(G; k), not a chromatic, and even not an edge elimination invariant. Finally, we study whether the Harary polynomials are definable in monadic second-order Logic. Keywords: Generalized colorings; Graph polynomials; Courcelle's Theorem 2020 Mathematics Subject Classification: 05; 05C30; 05C31 1. -
Excluding a Ladder
EXCLUDING A LADDER TONY HUYNH, GWENAEL¨ JORET, PIOTR MICEK, MICHALT. SEWERYN, AND PAUL WOLLAN Abstract. A ladder is a 2 × k grid graph. When does a graph class C exclude some ladder as a minor? We show that this is the case if and only if all graphs G in C admit a proper vertex coloring with a bounded number of colors such that for every 2-connected subgraph H of G, there is a color that appears exactly once in H. This type of vertex coloring is a relaxation of the notion of centered coloring, where for every connected subgraph H of G, there must be a color that appears exactly once in H. The minimum number of colors in a centered coloring of G is the treedepth of G, and it is known that classes of graphs with bounded treedepth are exactly those that exclude a fixed path as a subgraph, or equivalently, as a minor. In this sense, the structure of graphs excluding a fixed ladder as a minor resembles the structure of graphs without long paths. Another similarity is as follows: It is an easy observation that every connected graph with two vertex-disjoint paths of length k has a path of length k + 1. We show that every 3-connected graph which contains as a minor a union of sufficiently many vertex-disjoint copies of a 2 × k grid has a 2 × (k + 1) grid minor. Our structural results have applications to poset dimension. We show that posets whose cover graphs exclude a fixed ladder as a minor have bounded dimension. -
Prizes and Awards Session
PRIZES AND AWARDS SESSION Wednesday, July 12, 2021 9:00 AM EDT 2021 SIAM Annual Meeting July 19 – 23, 2021 Held in Virtual Format 1 Table of Contents AWM-SIAM Sonia Kovalevsky Lecture ................................................................................................... 3 George B. Dantzig Prize ............................................................................................................................. 5 George Pólya Prize for Mathematical Exposition .................................................................................... 7 George Pólya Prize in Applied Combinatorics ......................................................................................... 8 I.E. Block Community Lecture .................................................................................................................. 9 John von Neumann Prize ......................................................................................................................... 11 Lagrange Prize in Continuous Optimization .......................................................................................... 13 Ralph E. Kleinman Prize .......................................................................................................................... 15 SIAM Prize for Distinguished Service to the Profession ....................................................................... 17 SIAM Student Paper Prizes .................................................................................................................... -
Minor-Minimal Non-Projective Planar Graphs with an Internal 3-Separation
MINOR-MINIMAL NON-PROJECTIVE PLANAR GRAPHS WITH AN INTERNAL 3-SEPARATION A Thesis Presented to The Academic Faculty by Arash Asadi Shahmirzadi In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in Algorithms, Combinatorics, and Optimization School of Mathematics Georgia Institute of Technology December, 2012 MINOR-MINIMAL NON-PROJECTIVE PLANAR GRAPHS WITH AN INTERNAL 3-SEPARATION Approved by: Dr. Robin Thomas, Advisor Dr. William T. Trotter School of Mathematics School of Mathematics Georgia Institute of Technology Georgia Institute of Technology Dr. William Cook Dr. Xingxing Yu School of Industrial and Systems School of Mathematics Engineering Georgia Institute of Technology Georgia Institute of Technology Dr. Prasad Tetali Date Approved: December, 2012 School of Mathematics Georgia Institute of Technology To my parents, for educating me in what truly matters. iii TABLE OF CONTENTS DEDICATION .................................. iii LIST OF FIGURES .............................. v SUMMARY .................................... vi I INTRODUCTION ............................. 1 1.1 GraphTheoreticPreliminaries . 1 1.2 GraphsonSurfaces ........................... 4 1.3 Embedding and Excluding Subgraphs and Minors . 5 1.4 Non-Planar Extensions of Planar Graphs . 9 1.5 Application of the list of minor minimal non-projective planar graphs 10 1.6 Previous approaches for finding the list of minor minimal non-projective planargraphs .............................. 13 1.7 Minor Minimal Non-Projective Planar Graphs . .. 15 1.8 MainResults .............................. 16 1.9 OutlineoftheProof .......................... 19 II NON-c-PLANAR EXTENSIONS OF A c-DISK SYSTEM .... 23 2.1 Definitions and Preliminaries . 23 2.2 Usefullemmas.............................. 31 III SOME APPLICATIONS OF THE THEORY TO ROOTED GRAPHS ......................................... 48 IV OBSTRUCTIONS FOR c-, ac-, abc-PLANARITY ......... 66 4.1 Obstructions for c-planarity ..................... -
Planar Emulators Conjecture Is Nearly True for Cubic Graphs
Planar Emulators Conjecture Is Nearly True for Cubic Graphs Martin Derkaa, Petr Hlinˇen´yb,1 aDavid R. Cheriton School of Computer Science, University of Waterloo, 200 University Avenue West, Waterloo, ON, N2L 3G1, Canada bFaculty of Informatics, Masaryk University, Botanick´a68a, 602 00 Brno, Czech Republic Abstract We prove that a cubic nonprojective graph cannot have a finite planar emu- lator, unless it belongs to one of two very special cases (in which the answer is open). This shows that Fellows' planar emulator conjecture, disproved for general graphs by Rieck and Yamashita in 2008, is nearly true on cubic graphs, and might very well be true there definitely. Keywords: planar emulator; projective planar graph; graph minor 1. Introduction A graph G has a finite planar emulator H if H is a planar graph and there is a graph homomorphism ' : V (H) ! V (G) where ' is locally surjective, i.e. for every vertex v 2 V (H), the neighbours of v in H are mapped surjectively onto the neighbours of '(v) in G. We also say that such a G is planar-emulable. If we insist on ' being locally bijective, we get H a planar cover. The concept of planar emulators was proposed in 1985 by M. Fellows [6], and it tightly relates (although of independent origin) to the better known planar cover conjecture of Negami [11]. Fellows also raised the main question: What is the class of graphs with finite planar emulators? Soon later he conjectured that the class of planar-emulable graphs coincides with the class of graphs with finite planar covers (conjectured to be the class of projective graphs by Negami [11]| still open nowadays). -
Every Monotone Graph Property Is Testable∗
Every Monotone Graph Property is Testable∗ Noga Alon † Asaf Shapira ‡ Abstract A graph property is called monotone if it is closed under removal of edges and vertices. Many monotone graph properties are some of the most well-studied properties in graph theory, and the abstract family of all monotone graph properties was also extensively studied. Our main result in this paper is that any monotone graph property can be tested with one-sided error, and with query complexity depending only on . This result unifies several previous results in the area of property testing, and also implies the testability of well-studied graph properties that were previously not known to be testable. At the heart of the proof is an application of a variant of Szemer´edi’s Regularity Lemma. The main ideas behind this application may be useful in characterizing all testable graph properties, and in generally studying graph property testing. As a byproduct of our techniques we also obtain additional results in graph theory and property testing, which are of independent interest. One of these results is that the query complexity of testing testable graph properties with one-sided error may be arbitrarily large. Another result, which significantly extends previous results in extremal graph-theory, is that for any monotone graph property P, any graph that is -far from satisfying P, contains a subgraph of size depending on only, which does not satisfy P. Finally, we prove the following compactness statement: If a graph G is -far from satisfying a (possibly infinite) set of monotone graph properties P, then it is at least δP ()-far from satisfying one of the properties. -
Graph Properties and Hypergraph Colourings
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Discrete Mathematics 98 (1991) 81-93 81 North-Holland Graph properties and hypergraph colourings Jason I. Brown Department of Mathematics, York University, 4700 Keele Street, Toronto, Ont., Canada M3l lP3 Derek G. Corneil Department of Computer Science, University of Toronto, Toronto, Ont., Canada MSS IA4 Received 20 December 1988 Revised 13 June 1990 Abstract Brown, J.I., D.G. Corneil, Graph properties and hypergraph colourings, Discrete Mathematics 98 (1991) 81-93. Given a graph property P, graph G and integer k 20, a P k-colouring of G is a function Jr:V(G)+ (1,. ) k} such that the subgraph induced by each colour class has property P. When P is closed under induced subgraphs, we can construct a hypergraph HG such that G is P k-colourable iff Hg is k-colourable. This correlation enables us to derive interesting new results in hypergraph chromatic theory from a ‘graphic’ approach. In particular, we build vertex critical hypergraphs that are not edge critical, construct uniquely colourable hypergraphs with few edges and find graph-to-hypergraph transformations that preserve chromatic numbers. 1. Introduction A property P is a collection of graphs (closed under isomorphism) containing K, and K, (all graphs considered here are finite); P is hereditary iff P is closed under induced subgraphs and nontrivial iff P does not contain every graph. Any graph G E P is called a P-graph. Throughout, we shall only be interested in hereditary and nontrivial properties, and P will always denote such a property. -
Isomorphism and Embedding Problems for Infinite Limits of Scale
Isomorphism and Embedding Problems for Infinite Limits of Scale-Free Graphs Robert D. Kleinberg ∗ Jon M. Kleinberg y Abstract structure of finite PA graphs; in particular, we give a The study of random graphs has traditionally been characterization of the graphs H for which the expected dominated by the closely-related models (n; m), in number of subgraph embeddings of H in an n-node PA which a graph is sampled from the uniform distributionG graph remains bounded as n goes to infinity. on graphs with n vertices and m edges, and (n; p), in n G 1 Introduction which each of the 2 edges is sampled independently with probability p. Recen tly, however, there has been For decades, the study of random graphs has been dom- considerable interest in alternate random graph models inated by the closely-related models (n; m), in which designed to more closely approximate the properties of a graph is sampled from the uniformG distribution on complex real-world networks such as the Web graph, graphs with n vertices and m edges, and (n; p), in n G the Internet, and large social networks. Two of the most which each of the 2 edges is sampled independently well-studied of these are the closely related \preferential with probability p.The first was introduced by Erd}os attachment" and \copying" models, in which vertices and R´enyi in [16], the second by Gilbert in [19]. While arrive one-by-one in sequence and attach at random in these random graphs have remained a central object \rich-get-richer" fashion to d earlier vertices. -
Minors and Dimension
MINORS AND DIMENSION BARTOSZ WALCZAK Abstract. It has been known for 30 years that posets with bounded height and with cover graphs of bounded maximum degree have bounded dimension. Recently, Streib and Trotter proved that dimension is bounded for posets with bounded height and planar cover graphs, and Joret et al. proved that dimension is bounded for posets with bounded height and with cover graphs of bounded tree-width. In this paper, it is proved that posets of bounded height whose cover graphs exclude a fixed topological minor have bounded dimension. This generalizes all the aforementioned results and verifies a conjecture of Joret et al. The proof relies on the Robertson-Seymour and Grohe-Marx graph structure theorems. 1. Introduction In this paper, we are concerned with finite partially ordered sets, which we simply call posets. The dimension of a poset P is the minimum number of linear orders that form a realizer of P , that is, their intersection gives rise to P . The notion of dimension was introduced in 1941 by Dushnik and Miller [3] and since then has been one of the most extensively studied parameters in the combinatorics of posets. Much of this research has been focused on understanding when and why dimension is bounded, and this is also the focus of the current paper. The monograph [24] contains a comprehensive introduction to poset dimension theory. To some extent, dimension for posets behaves like chromatic number for graphs. There is a natural construction of a poset with dimension d, the standard example Sd (see Figure1), which plays a similar role to the complete graph Kd in the graph setting.