Mantel's Theorem Research Articles

Extremal numbers of disjoint triangles in r-partite graphs

For two graphs G and F, the extremal number of F in G, denoted by ex(G,F), is the maximum number of edges in a spanning subgraph of G not containing F as a subgraph. Determining ex(Kn,F) for a given graph F is a classical extremal problem in graph theory. In 1962, Erdős determined ex(Kn,kK3), which generalized Mantel's Theorem. On the other hand, in 1974, Bollobás, Erdős, and Straus determined ex(Kn1,n2,…,nr,Kt), which extended Turán's Theorem to complete multipartite graphs. In this paper, we determine ex(Kn1,n2,…,nr,kK3) for r≥4 and 10k−4≤n1+4k≤n2≤n3≤⋯≤nr.

Maxima of the Q-index of non-bipartite C3-free graphs

A classic result in extremal graph theory, known as Mantel's theorem, states that every non-bipartite graph of order n with size m>⌊n24⌋ contains a triangle. Lin, Ning and Wu [Comb. Probab. Comput. 30 (2021) 258-270] proved a spectral version of Mantel's theorem for given order n. Zhai and Shu [Discrete Math. 345 (2022) 112630] investigated a spectral version for fixed size m. In this paper, we prove Q-spectral versions of Mantel's theorem.

Regular Turán numbers and some Gan–Loh–Sudakov‐type problems

AbstractMotivated by a Gan–Loh–Sudakov‐type problem, we introduce the regular Turán numbers, a natural variation on the classical Turán numbers where we restrict ourselves to the class of regular graphs. Among other results, we prove a striking supersaturation version of Mantel's theorem in the case of a regular host graph of odd order. We also characterize the graphs for which the regular Turán numbers behave classically or otherwise.

A spectral version of Mantel's theorem

A classic result in extremal graph theory, known as Mantel's theorem, implies that every non-bipartite graph of order n with size m>⌊n2/4⌋ contains a triangle. Recently, by majority technique, Lin, Ning and Wu obtained a spectral version as follows: every non-bipartite graph G of size m with spectral radius ρ(G)≥m−1 contains a triangle unless G≅C5. In this paper, by using completely different techniques we show that every non-bipartite graph G of size m with ρ(G)≥ρ⁎(m) contains a triangle unless G≅SK2,m−12, where ρ⁎(m) is the largest root of x3−x2−(m−2)x+(m−3)=0 and SK2,m−12 is obtained by subdividing an edge of K2,m−12. This result implies both Mantel's theorem and Lin, Ning and Wu's result. Moreover, following Nikiforov's result, we also prove that every non-bipartite graph G with m≥26 and ρ(G)≥ρ(K1,m−1+e) contains a quadrilateral unless G≅K1,m−1+e.

A rainbow version of Mantel's Theorem

Mantel's Theorem from 1907 is one of the oldest results in graph theory: every simple $n$-vertex graph with more than $\frac{1}{4}n^2$ edges contains a triangle. The theorem has been generalized in many different ways, including other subgraphs, minimum degree conditions, etc. This article deals with a generalization to edge-colored multigraphs, which can be viewed as a union of simple graphs, each corresponding to an edge-color class. The case of two colors is the same as the original setting: Diwan and Mubayi proved that any two graphs $G_1$ and $G_2$ on the same set of $n$ vertices, each containing more than $\frac{1}{4}n^2$ edges, give rise to a triangle with one edge from $G_1$ and two edges from $G_2$. The situation is however different for three colors. Fix $\tau=\frac{4-\sqrt{7}}{9}$ and split the $n$ vertices into three sets $A$, $B$ and $C$, such that $|B|=|C|=\lfloor\tau n\rfloor$ and $|A|=n-|B|-|C|$. The graph $G_1$ contains all edges inside $A$ and inside $B$, the graph $G_2$ contains all edges inside $A$ and inside $C$, and the graph $G_3$ contains all edges between $A$ and $B\cup C$ and inside $B\cup C$. It is easy to check that there is no triangle with one edge from $G_1$, one from $G_2$ and one from $G_3$; each of the graphs has about $\frac{1+\tau^2}{4}n^2=\frac{26-2\sqrt{7}}{81}n^2\approx 0.25566n^2$ edges. The main result of the article is that this construction is optimal: any three graphs $G_1$, $G_2$ and $G_3$ on the same set of $n$ vertices, each containing more than $\frac{1+\tau^2}{4}n^2$ edges, give rise to a triangle with one edge from each of the graphs $G_1$, $G_2$ and $G_3$. A computer-assisted proof of the same bound has been found by Culver, Lidický, Pfender and Volec.

Packing parameters in graphs: new bounds and a solution to an open problem

In this paper, we investigate the packing parameters in graphs. By applying the Mantel's theorem, We give upper bounds on packing and open packing numbers of triangle-free graphs along with characterizing the graphs for which the equalities hold and exhibit sharp Nordhaus-Gaddum type inequalities for packing numbers. We also solve the open problem of characterizing all connected graphs with $\rho_{o}(G)=n-\omega(G)$ posed in [S. Hamid and S. Saravanakumar, {\em Packing parameters in graphs}, Discuss Math. Graph Theory, 35 (2015), 5--16].

Largest Minimal Inversion-Complete and Pair-Complete Sets of Permutations

We solve two related extremal problems in the theory of permutations. A set 𝑄 of permutations of the integers 1 to 𝑛 is inversion-complete (resp., pair-complete) if for every inversion (𝑗, 𝑖), where 1 ≤ 𝑖 < 𝑗 ≤ 𝑛, (resp., for every pair (𝑖, 𝑗), where 𝑖 ≠ 𝑗) there exists a permutation in 𝑄 where 𝑗 is before 𝑖. It is minimally inversion-complete if in addition no proper subset of 𝑄 is inversion-complete; and similarly for pair completeness. The problems we consider are to determine the maximum cardinality of a minimal inversion-complete set of permutations, and that of a minimal pair-complete set of permutations. The latter problem arises in the determination of the Caratheodory numbers for certain abstract convexity structures on the (𝑛 − 1)-dimensional real and integer vector spaces. Using Mantel's Theorem on the maximum number of edges in a triangle-free graph, we determine these two maximum cardinalities and we present a complete description of the optimal sets of permutations for each problem. Perhaps surprisingly (since there are twice as many pairs to cover as inversions), these two maximum cardinalities coincide whenever 𝑛 ≥ 4.

Mantel's theorem for random hypergraphs

A classical result in extremal graph theory is Mantel's Theorem, which states that every maximum triangle-free subgraph of Kn is bipartite. A sparse version of Mantel's Theorem is that, for sufficiently large p, every maximum triangle-free subgraph of G(n, p) is w.h.p. bipartite. Recently, DeMarco and Kahn proved this for p>Klogn/n for some constant K, and apart from the value of the constant this bound is best possible. We study an extremal problem of this type in random hypergraphs. Denote by F5, which is sometimes called the generalized triangle, the 3-uniform hypergraph with vertex set {a,b,c,d,e} and edge set {abc,ade,bde}. One of the first results in extremal hypergraph theory is by Frankl and Furedi, who proved that the maximum 3-uniform hypergraph on n vertices containing no copy of F5 is tripartite for n > 3000. A natural question is for what p is every maximum F5-free subhypergraph of G3(n,p) w.h.p. tripartite. We show this holds for p>Klogn/n for some constant K and does not hold for p=0.1logn/n. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 48, 641–654, 2016

An Exact Turán Result for Tripartite 3-Graphs

Mantel's theorem says that among all triangle-free graphs of a given order the balanced complete bipartite graph is the unique graph of maximum size. We prove an analogue of this result for 3-graphs. Let $K_4^-=\{123,124,134\}$, $F_6=\{123,124,345,156\}$ and $\mathcal{F}=\{K_4^-,F_6\}$: for $n\neq 5$ the unique $\mathcal{F}$-free 3-graph of order $n$ and maximum size is the balanced complete tripartite 3-graph $S_3(n)$ (for $n=5$ it is $C_5^{(3)}=\{123,234,345,145,125\}$). This extends an old result of Bollobás that $S_3(n) $ is the unique 3-graph of maximum size with no copy of $K_4^-=\{123,124,134\}$ or $F_5=\{123,124,345\}$.

A Density Corrádi–Hajnal Theorem

AbstractWe find, for all sufficiently largenand eachk, the maximum number of edges in ann-vertex graph that does not containk+ 1 vertex-disjoint triangles.This extends a result of Moon [Canad. J.Math. 20 (1968), 96–102], which is in turn an extension of Mantel's Theorem. Our result can also be viewed as a density version of the Corrádi–Hajnal Theorem.

Mantel's theorem for random graphs

ABSTRACTFor a graph G, denote by t(G) (resp. b(G)) the maximum size of a triangle‐free (resp. bipartite) subgraph of G. Of course for any G, and a classic result of Mantel from 1907 (the first case of Turán's Theorem) says that equality holds for complete graphs. A natural question, first considered by Babai, Simonovits and Spencer about 20 years ago is, when (i.e., for what p = p(n)) is the “Erdős‐Rényi” random graph G = G(n,p) likely to satisfy t(G) = b(G)? We show that this is true if for a suitable constant C, which is best possible up to the value of C. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 59–72, 2015

A density Corrádi-Hajnal theorem

We find, for all sufficiently large n and each k, the maximum number of edges in an n-vertex graph which does not contain k+1 vertex-disjoint triangles. This extends a result of Moon [Canad. J. Math. 20 (1968), 96-102] which is in turn an extension of Mantel's Theorem. Our result can also be viewed as a density version of the Corradi-Hajnal Theorem.

Quadruple systems with independent neighborhoods

A 4-graph is odd if its vertex set can be partitioned into two sets so that every edge intersects both parts in an odd number of points. Let b ( n ) = max α { α ( n − α 3 ) + ( n − α ) ( α 3 ) } = ( 1 2 + o ( 1 ) ) ( n 4 ) denote the maximum number of edges in an n-vertex odd 4-graph. Let n be sufficiently large, and let G be an n-vertex 4-graph such that for every triple xyz of vertices, the neighborhood N ( x y z ) = { w : w x y z ∈ G } is independent. We prove that the number of edges of G is at most b ( n ) . Equality holds only if G is odd with the maximum number of edges. We also prove that there is ε > 0 such that if the 4-graph G has minimum degree at least ( 1 / 2 − ε ) ( n 3 ) , then G is 2-colorable. Our results can be considered as a generalization of Mantel's theorem about triangle-free graphs, and we pose a conjecture about k-graphs for larger k as well.

A new generalization of Mantel's theorem to k-graphs

Let the k-graph Fan k consist of k edges that pairwise intersect exactly in one vertex x, plus one more edge intersecting each of these edges in a vertex different from x. We prove that, for n sufficiently large, the maximum number of edges in an n-vertex k-graph containing no copy of Fan k is ∏ i = 1 k ⌊ n + i − 1 k ⌋ , which equals the number of edges in a complete k-partite k-graph with almost equal parts. This is the only extremal example. This result is a special case of our more general theorem that applies to a larger class of excluded configurations.

The maximal number of edges in a homogeneous hypergraph containing no prohibited subgraphs

and, moreover, the extremal construction is unique to within isomorphism. Obtaining analogous results for s (i.e., families of s turned out to be much harder. Although this problem has received attention for a long time in combinatorial analysis (see [3]), there has been comparatively little progress. We can reformulate Mantel's theorem (bearing in mind the uniqueness of the extremal construction, proved by Turan) in the following form: THEOREM C [i, 2]. The maximal number of terms in a family of 2-subsets of an n-set, where no subset contains the symmetric difference of two others, is equal to