Notes on Extremal Graph Theory
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Avoiding Rainbow Induced Subgraphs in Vertex-Colorings
Avoiding rainbow induced subgraphs in vertex-colorings Maria Axenovich and Ryan Martin∗ Department of Mathematics, Iowa State University, Ames, IA 50011 [email protected], [email protected] Submitted: Feb 10, 2007; Accepted: Jan 4, 2008 Mathematics Subject Classification: 05C15, 05C55 Abstract For a fixed graph H on k vertices, and a graph G on at least k vertices, we write G −→ H if in any vertex-coloring of G with k colors, there is an induced subgraph isomorphic to H whose vertices have distinct colors. In other words, if G −→ H then a totally multicolored induced copy of H is unavoidable in any vertex-coloring of G with k colors. In this paper, we show that, with a few notable exceptions, for any graph H on k vertices and for any graph G which is not isomorphic to H, G 6−→ H. We explicitly describe all exceptional cases. This determines the induced vertex- anti-Ramsey number for all graphs and shows that totally multicolored induced subgraphs are, in most cases, easily avoidable. 1 Introduction Let G =(V, E) be a graph. Let c : V (G) → [k] be a vertex-coloring of G. We say that G is monochromatic under c if all vertices have the same color and we say that G is rainbow or totally multicolored under c if all vertices of G have distinct colors. The existence of a arXiv:1605.06172v1 [math.CO] 19 May 2016 graph forcing an induced monochromatic subgraph isomorphic to H is well known. The following bounds are due to Brown and R¨odl: Theorem 1 (Vertex-Induced Graph Ramsey Theorem [6]) For all graphs H, and all positive integers t there exists a graph Rt(H) such that if the vertices of Rt(H) are colored with t colors, then there is an induced subgraph of Rt(H) isomorphic to H which is monochromatic. -
Three Conjectures in Extremal Spectral Graph Theory
Three conjectures in extremal spectral graph theory Michael Tait and Josh Tobin June 6, 2016 Abstract We prove three conjectures regarding the maximization of spectral invariants over certain families of graphs. Our most difficult result is that the join of P2 and Pn−2 is the unique graph of maximum spectral radius over all planar graphs. This was conjectured by Boots and Royle in 1991 and independently by Cao and Vince in 1993. Similarly, we prove a conjecture of Cvetkovi´cand Rowlinson from 1990 stating that the unique outerplanar graph of maximum spectral radius is the join of a vertex and Pn−1. Finally, we prove a conjecture of Aouchiche et al from 2008 stating that a pineapple graph is the unique connected graph maximizing the spectral radius minus the average degree. To prove our theorems, we use the leading eigenvector of a purported extremal graph to deduce structural properties about that graph. Using this setup, we give short proofs of several old results: Mantel's Theorem, Stanley's edge bound and extensions, the K}ovari-S´os-Tur´anTheorem applied to ex (n; K2;t), and a partial solution to an old problem of Erd}oson making a triangle-free graph bipartite. 1 Introduction Questions in extremal graph theory ask to maximize or minimize a graph invariant over a fixed family of graphs. Perhaps the most well-studied problems in this area are Tur´an-type problems, which ask to maximize the number of edges in a graph which does not contain fixed forbidden subgraphs. Over a century old, a quintessential example of this kind of result is Mantel's theorem, which states that Kdn=2e;bn=2c is the unique graph maximizing the number of edges over all triangle-free graphs. -
Forbidding Subgraphs
Graph Theory and Additive Combinatorics Lecturer: Prof. Yufei Zhao 2 Forbidding subgraphs 2.1 Mantel’s theorem: forbidding a triangle We begin our discussion of extremal graph theory with the following basic question. Question 2.1. What is the maximum number of edges in an n-vertex graph that does not contain a triangle? Bipartite graphs are always triangle-free. A complete bipartite graph, where the vertex set is split equally into two parts (or differing by one vertex, in case n is odd), has n2/4 edges. Mantel’s theorem states that we cannot obtain a better bound: Theorem 2.2 (Mantel). Every triangle-free graph on n vertices has at W. Mantel, "Problem 28 (Solution by H. most bn2/4c edges. Gouwentak, W. Mantel, J. Teixeira de Mattes, F. Schuh and W. A. Wythoff). Wiskundige Opgaven 10, 60 —61, 1907. We will give two proofs of Theorem 2.2. Proof 1. G = (V E) n m Let , a triangle-free graph with vertices and x edges. Observe that for distinct x, y 2 V such that xy 2 E, x and y N(x) must not share neighbors by triangle-freeness. Therefore, d(x) + d(y) ≤ n, which implies that d(x)2 = (d(x) + d(y)) ≤ mn. ∑ ∑ N(y) x2V xy2E y On the other hand, by the handshake lemma, ∑x2V d(x) = 2m. Now by the Cauchy–Schwarz inequality and the equation above, Adjacent vertices have disjoint neigh- borhoods in a triangle-free graph. !2 ! 4m2 = ∑ d(x) ≤ n ∑ d(x)2 ≤ mn2; x2V x2V hence m ≤ n2/4. -
Extremal Graph Theory
Extremal Graph Theory Ajit A. Diwan Department of Computer Science and Engineering, I. I. T. Bombay. Email: [email protected] Basic Question • Let H be a fixed graph. • What is the maximum number of edges in a graph G with n vertices that does not contain H as a subgraph? • This number is denoted ex(n,H). • A graph G with n vertices and ex(n,H) edges that does not contain H is called an extremal graph for H. Mantel’s Theorem (1907) n2 ex(n, K ) 3 4 • The only extremal graph for a triangle is the complete bipartite graph with parts of nearly equal sizes. Complete Bipartite graph Turan’s theorem (1941) t 2 ex(n, K ) n2 t 2(t 1) • Equality holds when n is a multiple of t-1. • The only extremal graph is the complete (t-1)- partite graph with parts of nearly equal sizes. Complete Multipartite Graph Proofs of Turan’s theorem • Many different proofs. • Use different techniques. • Techniques useful in proving other results. • Algorithmic applications. • “BOOK” proofs. Induction • The result is trivial if n <= t-1. • Suppose n >= t and consider a graph G with maximum number of edges and no Kt. • G must contain a Kt-1. • Delete all vertices in Kt-1. • The remaining graph contains at most t 2 (n t 1)2 edges. 2(t 1) Induction • No vertex outside Kt-1 can be joined to all vertices of Kt-1. • Total number of edges is at most t 2 (t 1)(t 2) (n t 1)2 2(t 1) 2 t 2 (n t 1)(t 2) n2 2(t 1) Greedy algorithm • Consider any extremal graph and let v be a vertex with maximum degree ∆. -
An Extremal Problem for Complete Bipartite Graphs
Studia Seientiarum Mathematicarum Hungarica 23 (1988), 319-326 AN EXTREMAL PROBLEM FOR COMPLETE BIPARTITE GRAPHS P. ERDŐS, R. J. FAUDREE, C . C . ROUSSEAU and R . H. SCHELP Dedicated to the memory of Paul Turán Abstract Define f(n, k) to be the largest integer q such that for every graph G of order n and size q, G contains every complete bipartite graph K u, ,, with a+h=n-k . We obtain (i) exact values for f(n, 0) and f(n, 1), (ii) upper and lower bounds for f(n, k) when ku2 is fixed and n is large, and (iii) an upper bound for f(n, lenl) . 1 . Introduction Extremal graph theory, which was initiated by Turán in 1941 [4], is still the source of many interesting and difficult problems . The standard problem is to deter- mine f(n, G), the smallest integer q such that every graph with n vertices and q edges contains a subgraph isomorphic to G . It is striking that whereas Turán completely determined f(n, Km), there is much which is as yet unknown concerning f(n, Ka, b). In this paper, we consider a variant of the extremal problem for complete bipartite graphs . In this variant we ask how many edges must be deleted from K„ so that the resulting graph no longer contains Ka,, for some pair (a, b) with a+b=in . Specifically, we seek to determine an extremal function f(n, k) defined as follows . For in ::- 1, let B,,, denote the class of all graphs G such that GDKa , b for every pair (a, b) with a+b=m . -
The History of Degenerate (Bipartite) Extremal Graph Problems
The history of degenerate (bipartite) extremal graph problems Zolt´an F¨uredi and Mikl´os Simonovits May 15, 2013 Alfr´ed R´enyi Institute of Mathematics, Budapest, Hungary [email protected] and [email protected] Abstract This paper is a survey on Extremal Graph Theory, primarily fo- cusing on the case when one of the excluded graphs is bipartite. On one hand we give an introduction to this field and also describe many important results, methods, problems, and constructions. 1 Contents 1 Introduction 4 1.1 Some central theorems of the field . 5 1.2 Thestructureofthispaper . 6 1.3 Extremalproblems ........................ 8 1.4 Other types of extremal graph problems . 10 1.5 Historicalremarks . .. .. .. .. .. .. 11 arXiv:1306.5167v2 [math.CO] 29 Jun 2013 2 The general theory, classification 12 2.1 The importance of the Degenerate Case . 14 2.2 The asymmetric case of Excluded Bipartite graphs . 15 2.3 Reductions:Hostgraphs. 16 2.4 Excluding complete bipartite graphs . 17 2.5 Probabilistic lower bound . 18 1 Research supported in part by the Hungarian National Science Foundation OTKA 104343, and by the European Research Council Advanced Investigators Grant 267195 (ZF) and by the Hungarian National Science Foundation OTKA 101536, and by the European Research Council Advanced Investigators Grant 321104. (MS). 1 F¨uredi-Simonovits: Degenerate (bipartite) extremal graph problems 2 2.6 Classification of extremal problems . 21 2.7 General conjectures on bipartite graphs . 23 3 Excluding complete bipartite graphs 24 3.1 Bipartite C4-free graphs and the Zarankiewicz problem . 24 3.2 Finite Geometries and the C4-freegraphs . -
1 Turán's Theorem
Yuval Wigderson Turandotdotdot January 27, 2020 Ma il mio mistero `echiuso in me Nessun dorma, from Puccini's Turandot Libretto: G. Adami and R. Simoni 1 Tur´an'stheorem How many edges can we place among n vertices in such a way that we make no triangle? This is perhaps the first question asked in the field of extremal graph theory, which is the topic of this talk. Upon some experimentation, one can make the conjecture that the best thing to do is to split the vertices into two classes, of sizes x and n − x, and then connect any two vertices in different classes. This will not contain a triangle, since by the pigeonhole principle, among any three vertices there will be two in the same class, which will not be adjacent. This construction will have x(n − x) edges, and by the AM-GM inequality, we see that this is maximized when x and n − x are as close as possible; namely, our classes should n n n n n2 1 n have sizes b 2 c and d 2 e. Thus, this graph will have b 2 cd 2 e ≈ 4 ≈ 2 2 edges. Indeed, this is the unique optimal construction, as was proved by Mantel in 1907. In 1941, Tur´anconsidered a natural extension of this problem, where we forbid not a triangle, but instead a larger complete graph Kr+1. As before, a natural construction is to split the vertices into r almost-equal classes and connect all pairs of vertices in different classes. Definition 1. -
Strongly Regular Graphs
MT5821 Advanced Combinatorics 1 Strongly regular graphs We introduce the subject of strongly regular graphs, and the techniques used to study them, with two famous examples. 1.1 The Friendship Theorem This theorem was proved by Erdos,˝ Renyi´ and Sos´ in the 1960s. We assume that friendship is an irreflexive and symmetric relation on a set of individuals: that is, nobody is his or her own friend, and if A is B’s friend then B is A’s friend. (It may be doubtful if these assumptions are valid in the age of social media – but we are doing mathematics, not sociology.) Theorem 1.1 In a finite society with the property that any two individuals have a unique common friend, there must be somebody who is everybody else’s friend. In other words, the configuration of friendships (where each individual is rep- resented by a dot, and friends are joined by a line) is as shown in the figure below: PP · PP · u P · u · " " BB " u · " B " " B · "B" bb T b ub u · T b T b b · T b T · T u PP T PP u P u u 1 1.2 Graphs The mathematical model for a structure of the type described in the Friendship Theorem is a graph. A simple graph (one “without loops or multiple edges”) can be regarded as a set carrying an irreflexive and symmetric binary relation. The points of the set are called the vertices of the graph, and a pair of points which are related is called an edge. -
Extremal Graph Problems, Degenerate Extremal Problems, and Supersaturated Graphs
Extremal Graph Problems, Degenerate Extremal Problems, and Supersaturated Graphs Mikl´os Simonovits Aug, 2001 This is a LATEX version of my paper, from “Progress in Graph Theory”, the Waterloo Silver Jubilee Converence Proceedings, eds. Bondy and Murty, with slight changes in the notations, and some slight corrections. A Survey of Old and New Results Notation. Given a graph, hypergraph Gn, . , the upper index always de- notes the number of vertices, e(G), v(G) and χ(G) denote the number of edges, vertices and the chromatic number of G respectively. Given a family of graphs, hypergraphs, ex(n, ) denotes the maximum number of edges L L (hyperedges) a graph (hypergraph) Gn of order n can have without containing subgraphs (subhypergraphs) from . The problem of determining ex(n, ) is called a Tur´an-type extremal problemL . The graphs attaining the maximumL will be called extremal and their family will be denoted by EX(n, ). L 1 1. Introduction Let us restrict our consideration to ordinary graphs without loops and multi- ple edges. In 1940, P. Tur´an posed and solved the extremal problem of Kp+1, the complete graph on p + 1 vertices [39, 40]: ´ TURAN THEOREM. If Tn,p denotes the complete p–partite graph of order n having the maximum number of edges (or, in other words, the graph obtained by partitioning n vertices into p classes as equally as possible, and then joining two vertices iff they belong to different classes), then (a) If Tn,p contains no Kp+1, and (b) all the other graphs Gn of order n not containing Kp+1 have less than e(Tn,p) edges. -
Dynamic Cage Survey
Dynamic Cage Survey Geoffrey Exoo Department of Mathematics and Computer Science Indiana State University Terre Haute, IN 47809, U.S.A. [email protected] Robert Jajcay Department of Mathematics and Computer Science Indiana State University Terre Haute, IN 47809, U.S.A. [email protected] Department of Algebra Comenius University Bratislava, Slovakia [email protected] Submitted: May 22, 2008 Accepted: Sep 15, 2008 Version 1 published: Sep 29, 2008 (48 pages) Version 2 published: May 8, 2011 (54 pages) Version 3 published: July 26, 2013 (55 pages) Mathematics Subject Classifications: 05C35, 05C25 Abstract A(k; g)-cage is a k-regular graph of girth g of minimum order. In this survey, we present the results of over 50 years of searches for cages. We present the important theorems, list all the known cages, compile tables of current record holders, and describe in some detail most of the relevant constructions. the electronic journal of combinatorics (2013), #DS16 1 Contents 1 Origins of the Problem 3 2 Known Cages 6 2.1 Small Examples . 6 2.1.1 (3,5)-Cage: Petersen Graph . 7 2.1.2 (3,6)-Cage: Heawood Graph . 7 2.1.3 (3,7)-Cage: McGee Graph . 7 2.1.4 (3,8)-Cage: Tutte-Coxeter Graph . 8 2.1.5 (3,9)-Cages . 8 2.1.6 (3,10)-Cages . 9 2.1.7 (3,11)-Cage: Balaban Graph . 9 2.1.8 (3,12)-Cage: Benson Graph . 9 2.1.9 (4,5)-Cage: Robertson Graph . 9 2.1.10 (5,5)-Cages . -
Chapter 9 Introduction to Extremal Graph Theory
Chapter 9 Introduction to Extremal Graph Theory Prof. Tesler Math 154 Winter 2020 Prof. Tesler Ch. 9: Extremal Graph Theory Math 154 / Winter 2020 1 / 50 Avoiding a subgraph F G Let F and G be graphs. G is called F-free if there’s no subgraph isomorphic to F. An example is above. Prof. Tesler Ch. 9: Extremal Graph Theory Math 154 / Winter 2020 2 / 50 Avoiding a subgraph F G Is the graph on the right F-free? Prof. Tesler Ch. 9: Extremal Graph Theory Math 154 / Winter 2020 3 / 50 Avoiding a subgraph F G No. There are subgraphs isomorphic to F, even though they’re drawn differently than F. Prof. Tesler Ch. 9: Extremal Graph Theory Math 154 / Winter 2020 4 / 50 Extremal Number Question Given a graph F (to avoid), and a positive integer n, what’s the largest # of edges an F-free graph on n vertices can have? This number is denoted ex(n, F). This number is called the extremal number or Turán number of F. An F-free graph with n vertices and ex(n, F) edges is called an extremal graph. Prof. Tesler Ch. 9: Extremal Graph Theory Math 154 / Winter 2020 5 / 50 Extremal Number for K1,2 G F = K1,2 = Let F = P2 = K1,2. (A two edge path and K1,2 are the same.) For this F, being F-free means no vertex can be in > 2 edges. So, an F-free graph G must consist of vertex-disjoint edges (a matching) and/or isolated vertices. -
On Total Regularity of Mixed Graphs with Order Close to the Moore Bound
On total regularity of mixed graphs with order close to the Moore bound James Tuite∗ Grahame Erskine ∗ [email protected] [email protected] Abstract The undirected degree/diameter and degree/girth problems and their directed analogues have been studied for many decades in the search for efficient network topologies. Recently such questions have received much attention in the setting of mixed graphs, i.e. networks that admit both undirected edges and directed arcs. The degree/diameter problem for mixed graphs asks for the largest possible order of a network with diameter k, maximum undirected degree r and maximum directed out-degree z. It is also of interest to find smallest ≤ ≤ possible k-geodetic mixed graphs with minimum undirected degree r and minimum directed ≥ out-degree z. A simple counting argument reveals the existence of a natural bound, the ≥ Moore bound, on the order of such graphs; a graph that meets this limit is a mixed Moore graph. Mixed Moore graphs can exist only for k = 2 and even in this case it is known that they are extremely rare. It is therefore of interest to search for graphs with order one away from the Moore bound. Such graphs must be out-regular; a much more difficult question is whether they must be totally regular. For k = 2, we answer this question in the affirmative, thereby resolving an open problem stated in a recent paper of L´opez and Miret. We also present partial results for larger k. We finally put these results to practical use by proving the uniqueness of a 2-geodetic mixed graph with order exceeding the Moore bound by one.