An Extremal Property of Turán Graphs, II (original) (raw)
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An Extremal Property of Turán Graphs
Electronic Journal of Combinatorics, 2010
Let F n,tr(n) denote the family of all graphs on n vertices and t r (n) edges, where t r (n) is the number of edges in the Turán's graph T r (n)-the complete r-partite graph on n vertices with partition sizes as equal as possible. For a graph G and a positive integer λ, let P G (λ) denote the number of proper vertex colorings of G with at most λ colors, and let f (n, t r (n), λ) = max{P G (λ) : G ∈ F n,tr(n) }. We prove that for all n ≥ r ≥ 2, f (n, t r (n), r + 1) = P Tr(n) (r + 1) and that T r (n) is the only extremal graph.
Turán Numbers of Bipartite Graphs and Related Ramsey-Type Questions
Combinatorics, Probability and Computing, 2003
For a graph H and an integer n, the Turán number is the maximum possible number of edges in a simple graph on n vertices that contains no copy of H. H is r-degenerate if every one of its subgraphs contains a vertex of degree at most r. We prove that, for any fixed bipartite graph H in which all degrees in one colour class are at most r, . This is tight for all values of r and can also be derived from an earlier result of Füredi. We also show that there is an absolute positive constant c such that, for every fixed bipartite r-degenerate graph H, This is motivated by a conjecture of Erdős that asserts that, for every such H, For two graphs G and H, the Ramsey number is the minimum number n such that, in any colouring of the edges of the complete graph on n vertices by red and blue, there is either a red copy of G or a blue copy of H. Erdős conjectured that there is an absolute constant c such that, for any graph G with m edges, . Here we prove this conjecture for bipartite graphs G, a...
Another extremal problem for Turan graphs
Discrete Mathematics, 1987
We consider only finite, undirected graphs without loops or multiple edges. A clique of a graph G is a maximal complete subgraph of G. The clique number w(G) is the number of vertices in the largest clique of G. This note addresses the foflowing question: Which graphs G on n vertices with w(G) = r have the maximum number of cliques?
A Ramsey-type problem and the Turán numbers*
Journal of Graph Theory, 2002
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 k ≥ √ n + Ω(n 1/3), the largest value of m is asymptotically equal to the Turán number t(n, n 2 /k), while for any constant > 0, if k ≤ (1 −) √ n then the largest m is asymptotically larger than that Turán number.
Chromaticity of Turan Graphs with At Most Three Edges Deleted
Iranian Journal of Mathematical Sciences and Informatics, 2014
Let P(G,�) be the chromatic polynomial of a graph G. A graph G is chromatically unique if for any graph H, P(H,�) = P(G,�) implies H is isomorphic to G. In this paper, we determine the chro- maticity of all Turan graphs with at most three edges deleted. As a by product, we found many families of chromatically unique graphs and chromatic equivalence classes of graphs.
Turan numbers for bipartite graphs plus an odd cycle
2012
For an odd integer k, let C k = {C 3 , C 5 ,. .. , C k } denote the family of all odd cycles of length at most k and let C denote the family of all odd cycles. Erdős and Simonovits [10] conjectured that for every family F of bipartite graphs, there exists k such that ex n, F ∪ C k ∼ ex n, F ∪ C as n → ∞. This conjecture was proved by Erdős and Simonovits when F = {C 4 }, and for certain families of even cycles in [13]. In this paper, we give a general approach to the conjecture using Scott's sparse regularity lemma. Our approach proves the conjecture for complete bipartite graphs K 2,t and K 3,3 : we obtain more strongly that for any odd k ≥ 5, ex n, F ∪ {C k } ∼ ex n, F ∪ C and we show further that the extremal graphs can be made bipartite by deleting very few edges. In contrast, this formula does not extend to triangles-the case k = 3-and we give an algebraic construction for odd t ≥ 3 of K 2,t-free C 3-free graphs with substantially more edges than an extremal K 2,t-free bipartite graph on n vertices. Our general approach to the Erdős-Simonovits conjecture is effective based on some reasonable assumptions on the maximum number of edges in an m by n bipartite F-free graph.
Discrete Mathematics
We consider a natural generalisation of Turán's forbidden subgraph problem and the Ruzsa-Szemerédi problem by studying the maximum number exF (n, G) of edge-disjoint copies of a fixed graph F can be placed on an n-vertex ground set without forming a subgraph G whose edges are from different F-copies. We determine the pairs {F, G} for which the order of magnitude of exF (n, G) is quadratic and prove several asymptotic results using various tools from the regularity lemma and supersaturation to graph packing results.
Turán numbers and anti-Ramsey numbers for short cycles in complete 3-partite graphs
2020
We call a 4-cycle in K_n_1, n_2, n_3 multipartite, denoted by C_4^multi, if it contains at least one vertex in each part of K_n_1, n_2, n_3. The Turán number ex(K_n_1,n_2,n_3, C_4^multi) ( respectively, ex(K_n_1,n_2,n_3,{C_3, C_4^multi}) ) is the maximum number of edges in a graph G⊆ K_n_1,n_2,n_3 such that G contains no C_4^multi ( respectively, G contains neither C_3 nor C_4^multi ). We call a C^multi_4 rainbow if all four edges of it have different colors. The ant-Ramsey number ar(K_n_1,n_2,n_3, C_4^multi) is the maximum number of colors in an edge-colored of K_n_1,n_2,n_3 with no rainbow C_4^multi. In this paper, we determine that ex(K_n_1,n_2,n_3, C_4^multi)=n_1n_2+2n_3 and ar(K_n_1,n_2,n_3, C_4^multi)=ex(K_n_1,n_2,n_3, {C_3, C_4^multi})+1=n_1n_2+n_3+1, where n_1≥ n_2≥ n_3≥ 1.
Some exact results for generalized Turán problems
European Journal of Combinatorics, 2022
Fix a k-chromatic graph F . In this paper we consider the question to determine for which graphs H does the Turán graph T k-1 (n) have the maximum number of copies of H among all n-vertex F -free graphs (for n large enough). We say that such a graph H is F -Turán-good. In addition to some general results, we give (among others) the following concrete results: (i) For every complete multipartite graph H, there is k large enough such that H is K k -Turán-good. (ii) The path P 3 is F -Turán-good for F with χ(F ) ≥ 4. (iii) The path P 4 and cycle C 4 are C 5 -Turán-good. (iv) The cycle C 4 is F 2 -Turán-good where F 2 is the graph of two triangles sharing exactly one vertex.
On Turán numbers of the complete 4-graphs
Discrete Mathematics, 2021
The Turán number T (n, α+1, r) is the minimum number of edges in an n-vertex r-graph whose independence number does not exceed α. For each r ≥ 2, there exists t * (r) such that T (n, α + 1, r) = t * (r) n r α 1−r (1 + o(1)) as α/r → ∞ and n/α → ∞. It is known that t * (2) = 1/2, and the conjectured value of t * (3) is 2/3. We prove that t * (4) < 0.706335 .