A generalized Turán problem in random graphs

We study a generalization of the Turán problem in random graphs. Given graphs T and H, let ex(G(n,p),T,H) be the largest number of copies of T in an H‐free subgraph of G(n,p). We study the threshold phenomena arising in the evolution of the typical value of this random variable, for every H and every 2‐balanced T. Our results in the case when m₂(H) > m₂(T) are a natural generalization of the Erd?s‐Stone theorem for G(n,p), proved several years ago by Conlon‐Gowers and Schacht; the case T = K_m was previously resolved by Alon, Kostochka, and Shikhelman. The case when m₂(H) ≤ m₂(T) exhibits a more complex behavior. Here, the location(s) of the (possibly multiple) threshold(s) are determined by densities of various coverings of H with copies of T and the typical value(s) of ex(G(n,p),T,H) are given by solutions to deterministic hypergraph Turán‐type problems that we are unable to solve in full generality.     

Turán problems for integer‐weighted graphs

A multigraph is (k,r)‐dense if every k‐set spans at most r edges. What is the maximum number of edges ex_?(n,k,r) in a (k,r)‐dense multigraph on n vertices? We determine the maximum possible weight of such graphs for almost all k and r (e.g., for all r>k³) by determining a constant m=m(k,r) and showing that ex_?(n,k,r)=m +O(n), thus giving a generalization of Turán's theorem. We find exact answers in many cases, even when negative integer weights are also allowed. In fact, our main result is to determine the maximum weight of (k,r)‐dense n‐vertex multigraphs with arbitrary integer weights with an O(n) error term. © 2002 Wiley Periodicals, Inc. J Graph Theory 40: 195–225, 2002     

Triangles in C 5-free graphs and hypergraphs of girth six

We introduce a new approach and prove that the maximum number of triangles in a $C_5$-free graph on $n$ vertices is at most $$\left(1+o\left(1\right)\right)\frac{1}{3\sqrt{2}}{n}^{3∕2}.$$ We show a connection to $r$-uniform hypergraphs without (Berge) cycles of length less than six, and estimate their maximum possible size. Using our approach, we also (slightly) improve the previous estimate on the maximum size of an induced-$C_4$-free and $C_5$-free graph.     

Turánnical hypergraphs

This paper is motivated by the question of how global and dense restriction sets in results from extremal combinatorics can be replaced by less global and sparser ones. The result we consider here as an example is Turán's theorem, which deals with graphs G = ([n],E) such that no member of the restriction set \begin{align*}\mathcal {R}\end{align*} = \begin{align*}\left( {\begin{array}{*{20}c} {[n]} \\ r \\ \end{array} } \right)\end{align*} induces a copy of K_r. Firstly, we examine what happens when this restriction set is replaced by \begin{align*}\mathcal {R}\end{align*} = {X∈ \begin{align*}\left( {\begin{array}{*{20}c} {[n]} \\ r \\ \end{array} } \right)\end{align*}: X ∩ [m]≠??}. That is, we determine the maximal number of edges in an n ‐vertex such that no K_r hits a given vertex set. Secondly, we consider sparse random restriction sets. An r ‐uniform hypergraph \begin{align*}\mathcal R\end{align*} on vertex set [n] is called Turánnical (respectively ε ‐Turánnical), if for any graph G on [n] with more edges than the Turán number t_r(n) (respectively (1 + ε)t_r(n) ), no hyperedge of \begin{align*}\mathcal {R}\end{align*} induces a copy of K_r in G. We determine the thresholds for random r ‐uniform hypergraphs to be Turánnical and to be ε ‐Turánnical. Thirdly, we transfer this result to sparse random graphs, using techniques recently developed by Schacht [Extremal results for random discrete structures] to prove the Kohayakawa‐?uczak‐Rödl Conjecture on Turán's theorem in random graphs.© 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2012     

A Density Turán Theorem

Let F be a graph that contains an edge whose deletion reduces its chromatic number. For such a graph F , a classical result of Simonovits from 1966 shows that every graph on vertices with more than edges contains a copy of F . In this article we derive a similar theorem for multipartite graphs. For a graph H and an integer , let be the minimum real number such that every ?‐partite graph whose edge density between any two parts is greater than contains a copy of H . Our main contribution in this article is to show that for all sufficiently large if and only if H admits a vertex‐coloring with colors such that all color classes but one are independent sets, and the exceptional class induces just a matching. When H is a complete graph, this recovers a result of Pfender (Combinatorica 32 (2012), 483–495). We also consider several extensions of Pfender's result.     

A generalization of Turán's theorem

In this paper, we obtain an asymptotic generalization of Turán's theorem. We prove that if all the non‐trivial eigenvalues of a d‐regular graph G on n vertices are sufficiently small, then the largest K_t‐free subgraph of G contains approximately (t ? 2)/(t ? 1)‐fraction of its edges. Turán's theorem corresponds to the case d = n ? 1. © 2005 Wiley Periodicals, Inc. J Graph Theory     

On splittable colorings of graphs and hypergraphs

The notion of a split coloring of a complete graph was introduced by Erd?s and Gyárfás [ 7 ] as a generalization of split graphs. In this work, we offer an alternate interpretation by comparing such a coloring to the classical Ramsey coloring problem via a two‐round game played against an adversary. We show that the techniques used and bounds obtained on the extremal (r,m)‐split coloring problem of [ 7 ] are closer in nature to the Turán theory of graphs rather than Ramsey theory. We extend the notion of these colorings to hypergraphs and provide bounds and some exact results. © 2002 Wiley Periodicals, Inc. J Graph Theory 40: 226–237, 2002     

Extremal graphs with bounded densities of small subgraphs

Let Ex(n, k, μ) denote the maximum number of edges of an n-vertex graph in which every subgraph of k vertices has at most μ edges. Here we summarize some known results of the problem of determining Ex(n, k, μ), give simple proofs, and find some new estimates and extremal graphs. Besides proving new results, one of our main aims is to show how the classical Turáan theory can be applied to such problems. The case μ = is the famous result of Turáan. © 1998 John Wiley & Sons, Inc. J. Graph Theory 29: 185–207, 1998     

An application of the Turán theorem to domination in graphs

A function f:V(G)→{+1,−1} defined on the vertices of a graph G is a signed dominating function if for any vertex v the sum of function values over its closed neighborhood is at least 1. The signed domination number γ_s(G) of G is the minimum weight of a signed dominating function on G. By simply changing “{+1,−1}” in the above definition to “{+1,0,−1}”, we can define the minus dominating function and the minus domination number of G. In this note, by applying the Turán theorem, we present sharp lower bounds on the signed domination number for a graph containing no (k+1)-cliques. As a result, we generalize a previous result due to Kang et al. on the minus domination number of k-partite graphs to graphs containing no (k+1)-cliques and characterize the extremal graphs.     

On Turán problems for Cartesian products of graphs

Let be disjoint sets of sizes and . Let be a family of quadruples, having elements from and from , such that any subset with and contains one of the quadruples. We prove that the smallest size of is as . We also solve asymptotically a more general two‐partite Turán problem for quadruples.     

Stability results for random discrete structures

Two years ago, Conlon and Gowers, and Schacht proved general theorems that allow one to transfer a large class of extremal combinatorial results from the deterministic to the probabilistic setting. Even though the two papers solve the same set of long‐standing open problems in probabilistic combinatorics, the methods used in them vary significantly and therefore yield results that are not comparable in certain aspects. In particular, the theorem of Schacht yields stronger probability estimates, whereas the one of Conlon and Gowers also implies random versions of some structural statements such as the famous stability theorem of Erd?s and Simonovits. In this paper, we bridge the gap between these two transference theorems. Building on the approach of Schacht, we prove a general theorem that allows one to transfer deterministic stability results to the probabilistic setting. We then use this theorem to derive several new results, among them a random version of the Erd?s‐Simonovits stability theorem for arbitrary graphs, extending the result of Conlon and Gowers, who proved such a statement for so‐called strictly 2‐balanced graphs. The main new idea, a refined approach to multiple exposure when considering subsets of binomial random sets, may be of independent interest.Copyright © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 44, 269‐289, 2014     

Decomposing C4-free graphs under degree constraints

A celebrated theorem of Stiebitz 13 asserts that any graph with minimum degree at least can be partitioned into two parts that induce two subgraphs with minimum degree at least s and t, respectively. This resolved a conjecture of Thomassen. In this article, we prove that for , if a graph G contains no cycle of length four and has minimum degree at least , then G can be partitioned into two parts that induce two subgraphs with minimum degree at least s and t, respectively. This improves the result of Diwan in 5, where he proved the same statement for graphs of girth at least five. Our proof also works for the case of variable functions, in which the bounds are sharp as showing by some polarity graphs.     

Extremal graphs for the k‐flower

The Turán number of a graph H, , is the maximum number of edges in any graph of order n that does not contain an H as a subgraph. A graph on vertices consisting of k triangles that intersect in exactly one common vertex is called a k‐fan, and a graph consisting of k cycles that intersect in exactly one common vertex is called a k‐flower. In this article, we determine the Turán number of any k‐flower containing at least one odd cycle and characterize all extremal graphs provided n is sufficiently large. Erdős, Füredi, Gould, and Gunderson determined the Turán number for the k‐fan. Our result is a generalization of their result. The addition aim of this article is to draw attention to a powerful tool, the so‐called progressive induction lemma of Simonovits.     

On the local density problem for graphs of given odd-girth

Erdős conjectured that every n-vertex triangle-free graph contains a subset of vertices that spans at most edges. Extending a recent result of Norin and Yepremyan, we confirm this conjecture for graphs homomorphic to so-called Andrásfai graphs. As a consequence, Erdős' conjecture holds for every triangle-free graph G with minimum degree and if the degree condition can be relaxed to . In fact, we obtain a more general result for graphs of higher odd-girth.     

Regularity inheritance in pseudorandom graphs

Advancing the sparse regularity method, we prove one‐sided and two‐sided regularity inheritance lemmas for subgraphs of bijumbled graphs, improving on results of Conlon, Fox, and Zhao. These inheritance lemmas also imply improved H‐counting lemmas for subgraphs of bijumbled graphs, for some H.     

Maximum number of colorings of (2k,k2)‐graphs

Let consist of all simple graphs on 2k vertices and edges. For a simple graph G and a positive integer , let denote the number of proper vertex colorings of G in at most colors, and let . We prove that and is the only extremal graph. We also prove that as . © 2007 Wiley Periodicals, Inc. J Graph Theory 56: 135–148, 2007     

Generating cycles in graphs with at most one end

Answering a question of Halin, we prove that in a 3‐connected graph with at most one end the cycle space is generated by induced non‐separating cycles. © 2003 Wiley Periodicals, Inc. J Graph Theory 42: 342–349, 2003     

An application of Tutte’s Theorem to 1-factorization of regular graphs of high degree

A well known conjecture in graph theory states that every regular graph of even order 2n and degree λ(2n), where λ≥1/2, is 1-factorizable. Chetwynd and Hilton [A.G. Chetwynd, A.J.W. Hilton, 1-factorizing regular graphs of high degree — An improved bound, Discrete Math. 75 (1989) 103-112] and, independently, Niessen and Volkmann [T. Niessen, L. Volkmann, Class 1 conditions depending on the minimum degree and the number of vertices of maximum degree, J. Graph Theory (2) 14 (1990) 225-246] proved the above conjecture under the assumption that . Since these results were published no significant or even partial improvement has been made in terms of lowering the bound on λ. We shall obtain here a partial improvement on λ. Specifically, using the original Chetwynd-Hilton approach and Tutte’s 1-Factor Theorem, we show that the above bound can be improved to , apart (possibly) from two families of exceptional cases. We then show, under the stronger assumption that λ≥λ^∗≈0.785, that one of the two families of exceptional cases cannot occur.