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.     

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.     

Bounds on Turán determinants

Let μ denote a symmetric probability measure on [−1,1] and let (p_n) be the corresponding orthogonal polynomials normalized such that p_n(1)=1. We prove that the normalized Turán determinant Δ_n(x)/(1−x²), where , is a Turán determinant of order n−1 for orthogonal polynomials with respect to . We use this to prove lower and upper bounds for the normalized Turán determinant in the interval −1<x<1.     

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     

A Ramsey‐type problem and the Turán numbers*

For each n and k, we examine bounds on the largest number m so that for any k‐coloring of the edges of K_n there exists a copy of K_m whose edges receive at most k?1 colors. We show that for , the largest value of m is asymptotically equal to the Turá number , while for any constant then the largest m is asymptotically larger than that Turá number. © 2002 Wiley Periodicals, Inc. J Graph Theory 40: 120–129, 2002     

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 edge domination numbers of graphs

Let and be the signed edge domination number and signed star domination number of G, respectively. We prove that holds for all graphs G without isolated vertices, where n=|V(G)|?4 and m=|E(G)|, and pose some problems and conjectures.     

Lagrangian densities of linear forests and Turán numbers of their extensions

The Lagrangian of a hypergraph has been a useful tool in hypergraph extremal problems. Recently, Lagrangian densities of hypergraphs and Turán numbers of their extensions have been studied actively. However, determining the Lagrangian density of a hypergraph is not an easy task even for a “simple” hypergraph. For example, to determine the Lagrangian density of $K_4^3$ is equivalent to determine the Turán density of $K_4^3$ (a long standing conjecture of Turán). Hefetz and Keevash studied the Lagrangian density of the 3‐uniform matching of size 2. Pikhurko determined the Lagrangian density of a 4‐uniform tight path of length 2 and this led to confirm the conjecture of Frankl and Füredi on the Turán number of the $r$‐uniform generalized triangle for the case $r=4$. It is natural and interesting to consider Lagrangian densities of other “basic” hypergraphs. In this paper, we determine the Lagrangian densities for a class of 3‐uniform linear forests. For positive integers $s$ and $t$, let $P_{s,t}$ be the disjoint union of a 3‐uniform linear path of length $s$ and $t$ pairwise disjoint edges. In this paper, we determine the Lagrangian densities of $P_{s,t}$ for any $t$ and $s=2$ or 3. Applying a modified version of Pikhurko's transference argument used by Brandt, Irwin, and Jiang, we obtain the Turán numbers of their extensions.     

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 comment to: Two classes of edge domination in graphs

Let G be a graph of order n and denote the signed edge domination number of G. In [B. Xu, Two classes of edge domination in graphs, Discrete Appl. Math. 154 (2006) 1541-1546] it was proved that for any graph G of order n, . But the method given in the proof is not correct. In this paper we give an example for which the method of proof given in [1] does not work.     

An Extremal Property of Turán Graphs,II

Let denote Turán's graph—the complete 2‐partite graph on n vertices with partition sizes as equal as possible. We show that for all , the graph has more proper vertex colorings in at most 4 colors than any other graph with the same number of vertices and edges.     

Two classes of edge domination in graphs

Let (, resp.) be the number of (local) signed edge domination of a graph G [B. Xu, On signed edge domination numbers of graphs, Discrete Math. 239 (2001) 179-189]. In this paper, we prove mainly that and hold for any graph G of order n(n?4), and pose several open problems and conjectures.     

A note on the signed edge domination number in graphs

Let G=(V(G),E(G)) be a graph. A function f:E(G)→{+1,−1} is called the signed edge domination function (SEDF) of G if ∑_e^′N[e]f(e^′)≥1 for every eE(G). The signed edge domination number of G is defined as is a SEDF of G}. Xu [Baogen Xu, Two classes of edge domination in graphs, Discrete Applied Mathematics 154 (2006) 1541–1546] researched on the edge domination in graphs and proved that for any graph G of order n(n≥4). In the article, he conjectured that: For any 2-connected graph G of order n(n≥2), . In this note, we present some counterexamples to the above conjecture and prove that there exists a family of k-connected graphs G_m,k with .     

On signed cycle domination in graphs

Let G=(V,E) be a graph, a function f:E→{−1,1} is said to be an signed cycle dominating function (SCDF) of G if ∑_e∈E(C)f(e)≥1 holds for any induced cycle C of G. The signed cycle domination number of G is defined as is an SCDF of G}. In this paper, we obtain bounds on , characterize all connected graphs G with , and determine the exact value of for some special classes of graphs G. In addition, we pose some open problems and conjectures.     

Lower bounds on the signed domination numbers of directed graphs

A numerical invariant of directed graphs concerning domination which is named signed domination number γ_S is studied in this paper. We present some sharp lower bounds for γ_S in terms of the order, the maximum degree and the chromatic number of a directed graph.     

The signed star domination numbers of the Cartesian product graphs

Let G be a graph with vertex set V(G) and edge set E(G). A function f:E(G)→{-1,1} is said to be a signed star dominating function of G if for every v∈V(G), where E_G(v)={uv∈E(G)|u∈V(G)}. The minimum of the values of , taken over all signed star dominating functions f on G, is called the signed star domination number of G and is denoted by γ_SS(G). In this paper, a sharp upper bound of γ_SS(G×H) is presented.     

Total minus domination in k-partite graphs

A function f defined on the vertices of a graph G=(V,E),f:V→{-1,0,1} is a total minus dominating function (TMDF) if the sum of its values over any open neighborhood is at least one. The weight of a TMDF is the sum of its function values over all vertices. The total minus domination number, denoted by , of G is the minimum weight of a TMDF on G. In this paper, a sharp lower bound on of k-partite graphs is given.