Maximum Number Of Edges Research Articles

Planar Turán number of intersecting triangles

The planar Turán number of a given graph H, denoted by exP(n,H), is the maximum number of edges over all planar graphs on n vertices that do not contain a copy of H as a subgraph. Let Hk be a friendship graph, which is obtained from k triangles by sharing a vertex. In this paper, we obtain sharp bounds of exP(n,Hk) and exP(n,K1+Pk+1) for all non-trivial cases, which improve the corresponding results of Lan, Shi and Song in ((2019) [9]).

Inverse Turán numbers

For given graphs G and F, the Turán number ex(G,F) is defined to be the maximum number of edges in an F-free subgraph of G. Foucaud, Krivelevich and Perarnau and later independently Briggs and Cox introduced a dual version of this problem wherein for a given number k, one maximizes the number of edges in a host graph G for which ex(G,H)<k.Addressing a problem of Briggs and Cox, we determine the asymptotic value of the inverse Turán number of the paths of length 4 and 5 and provide an improved lower bound for all paths of even length. Moreover, we obtain bounds on the inverse Turán number of even cycles giving improved bounds on the leading coefficient in the case of C4. Finally, we give multiple conjectures concerning the asymptotic value of the inverse Turán number of C4 and Pℓ, suggesting that in the latter problem the asymptotic behavior depends heavily on the parity of ℓ.

The Exact Linear Turán Number of the Sail

A hypergraph is linear if any two of its edges intersect in at most one vertex. The sail (or $3$-fan) $F^3$ is the $3$-uniform linear hypergraph consisting of $3$ edges $f_1, f_2, f_3$ pairwise intersecting in the same vertex $v$ and an additional edge $g$ intersecting each $f_i$ in a vertex different from $v$. The linear Turán number $\mathrm{ex}_{\mathrm{lin}}(n, F^3)$ is the maximum number of edges in a $3$-uniform linear hypergraph on $n$ vertices that does not contain a copy of $F^3$.&#x0D; Füredi and Gyárfás proved that if $n = 3k$, then $\mathrm{ex}_{\mathrm{lin}}(n, F^3) = k^2$ and the only extremal hypergraphs in this case are transversal designs. They also showed that if $n = 3k+2$, then $\mathrm{ex}_{\mathrm{lin}}(n, F^3) = k^2+k$, and the only extremal hypergraphs are truncated designs (which are obtained from a transversal design on $3k+3$ vertices with $3$ groups by removing one vertex and all the hyperedges containing it) along with three other small hypergraphs. However, the case when $n =3k+1$ was left open.&#x0D; In this paper, we solve this remaining case by proving that $\mathrm{ex}_{\mathrm{lin}}(n, F^3) = k^2+1$ if $n = 3k+1$, answering a question of Füredi and Gyárfás. We also characterize all the extremal hypergraphs. The difficulty of this case is due to the fact that these extremal examples are rather non-standard. In particular, they are not derived from transversal designs like in the other cases.

On the Turán numbers of [formula omitted] in [formula omitted]-partite graphs

Given graphs G and H, the Turán numberex(G,H)ofHinG is the maximum number of edges in a subgraph of G that contains no H. Chen et al. determined ex(Kϱ1,ϱ2,kK2) for all 1≤k≤ϱ1≤ϱ2. De Silva et al. determined ex(Kϱ1,…,ϱr,kKr) for all r≥2 and 1≤k≤ϱ1≤⋯≤ϱr. Moreover, De Silva et al. proposed an interesting generalization of ex(Kϱ1,…,ϱr,kKr): Determine ex(Kϱ1,…,ϱℓ,kKr) for ℓ≥r. In this paper, we give a proof of ex(Kϱ1,…,ϱℓ,kK2)=(k−1)∑i=2ℓϱi for all ℓ≥2 and 1≤k≤ϱ1≤⋯≤ϱℓ. We also determine the Turán numbers ex(Kϱ1,ϱ2,ϱ3,ϱ4,kK3) for all k≥1 and ϱ4≥ϱ3≥ϱ2≥ϱ1≥4(k−1), which gives a positive solution to a problem due to De Silva et al.

Extremal graphs for edge blow-up of graphs

Given a graph H and an integer p, the edge blow-up Hp+1 of H is the graph obtained from replacing each edge in H by a clique of order p+1 where the new vertices of the cliques are all distinct. The Turán numbers for edge blow-up of matchings were first studied by Erdős and Moon. In this paper, we determine the range of the Turán numbers for edge blow-up of all bipartite graphs and the exact Turán numbers for edge blow-up of all non-bipartite graphs. In addition, we characterize the extremal graphs for edge blow-up of all non-bipartite graphs. Our results also extend several known results, including the Turán numbers for edge blow-up of stars, paths and cycles. The method we used can also be applied to find a family of counter-examples to a conjecture posed by Keevash and Sudakov in 2004 concerning the maximum number of edges not contained in any monochromatic copy of H in a 2-edge-coloring of Kn.

The Turán number of the square of a path

The Turán number of a graph H, ex(n,H), is the maximum number of edges in a graph on n vertices which does not have H as a subgraph. Let Pk be the path with k vertices, the square Pk2 of Pk is obtained by joining the pairs of vertices with distance one or two in Pk. The powerful theorem of Erdős, Stone and Simonovits determines the asymptotic behavior of ex(n,Pk2). In the present paper, we determine the exact value of ex(n,P52) and ex(n,P62) and pose a conjecture for the exact value of ex(n,Pk2).

The spectral radius of graphs with no odd wheels

The odd wheel W2k+1 is the graph formed by joining a vertex to a cycle of length 2k. In this paper, we investigate the largest value of the spectral radius of the adjacency matrix of an n-vertex graph that does not contain W2k+1. We determine the structure of the spectral extremal graphs for all k≥2,k⁄∈{4,5}. When k=2, we show that these spectral extremal graphs are among the Turán-extremal graphs on n vertices that do not contain W2k+1 and have the maximum number of edges, but when k≥9, we show that the family of spectral extremal graphs and the family of Turán-extremal graphs are disjoint.

A note on the Turán number of disjoint union of wheels

The Turán number of a graph H, ex(n,H), is the maximum number of edges in a graph on n vertices which does not have H as a subgraph. A wheel Wn is an n-vertex graph formed by connecting a single vertex to all vertices of a cycle Cn−1. Let mW2k+1 (k≥3) denote the graph defined by taking m vertex disjoint copies of W2k+1. For sufficiently large n, we determine the Turán number and all extremal graphs for mW2k+1 (k≥3). Let Wh be the family of graphs obtain by the disjoint union of a finite number of wheels, such that, the number of even wheels in the union is h, (h≥1). For any W∈Wh, we also provide the Turán number and all extremal graphs for W, when n is sufficiently large.

On plane subgraphs of complete topological drawings

Topological drawings are representations of graphs in the plane, where vertices are represented by points, and edges by simple curves connecting the points. A drawing is simple if two edges intersect at most in a single point, either at a common endpoint or at a proper crossing. In this paper we study properties of maximal plane subgraphs of simple drawings D n of the complete graph K n on n vertices. Our main structural result is that maximal plane subgraphs are 2 -connected and what we call essentially 3 -edge-connected . Besides, any maximal plane subgraph contains at least ⌈3 n /2⌉ edges. We also address the problem of obtaining a plane subgraph of D n with the maximum number of edges, proving that this problem is NP-complete. However, given a plane spanning connected subgraph of D n , a maximum plane augmentation of this subgraph can be found in O ( n 3 ) time. As a side result, we also show that the problem of finding a largest compatible plane straight-line graph of two labeled point sets is NP-complete.

On the cover Turán number of Berge hypergraphs

For a fixed set of positive integers R, we say H is an R-uniform hypergraph, or R-graph, if the cardinality of each edge belongs to R. For a graph G=(V,E), a hypergraph H is called a Berge-G, if there is an injection i:V(G)→V(H) and a bijection f:E(G)→E(H) such that for all e=uv∈E(G), we have {i(u),i(v)}⊆f(e). We define the cover Turán number of a graph G, denoted as exˆR(n,BG), as the maximum number of edges in the 2-shadow of an n-vertex Berge-G-free R-graph, where the 2-shadow H of a hypergraph H is a graph such that an edge e∈E(H) if and only if e⊆h for some h∈E(H). In this paper, we show an Erdős–Stone–Simonovits-type upper bound on the cover Turán number of graphs and determine the cover Turán density of all graphs when the host hypergraph is 3-uniform.

Relative Turán numbers for hypergraph cycles

For an r-uniform hypergraph H and a family of r-uniform hypergraphs F, the relative Turán number ex(H,F) is the maximum number of edges in an F-free subgraph of H. In this paper we give lower bounds on ex(H,F) for certain families of hypergraph cycles F such as Berge cycles and loose cycles. In particular, if Cℓ3 denotes the set of all 3-uniform Berge ℓ-cycles and H is a 3-uniform hypergraph with maximum degree Δ, we proveex(H,C43)≥Δ−3/4−o(1)e(H),ex(H,C53)≥Δ−3/4−o(1)e(H), and these bounds are tight up to the o(1) term.

Turán numbers for disjoint paths

AbstractThe Turán number of a graph , ex, is the maximum number of edges in a graph of order that does not contain a copy of as a subgraph. Lidický, Liu and Palmer determined ex for sufficiently large and proved that the extremal graph is unique, where is the union of pairwise vertex‐disjoint paths on vertices. There are a few kinds of graphs for which ex are known exactly for all , including cliques, matchings, paths, cycles on odd number of vertices and some other special graphs. In this paper, using a different approach from Lidický, Liu and Palmer, we determine ex for all integers when at most one of is odd. Furthermore, we show that there exists a family of pairs of bipartite graphs such that for all integers , which is related to an old problem of Erdős and Simonovits.

On the Maximal Colorings of Complete Graphs Without Some Small Properly Colored Subgraphs

Let mathrm{pr}(K_{n}, G) be the maximum number of colors in an edge-coloring of K_{n} with no properly colored copy of G. For a family {mathcal {F}} of graphs, let mathrm{ex}(n, {mathcal {F}}) be the maximum number of edges in a graph G on n vertices which does not contain any graphs in {mathcal {F}} as subgraphs. In this paper, we show that mathrm{pr}(K_{n}, G)-mathrm{ex}(n, mathcal {G'})=o(n^{2}), where mathcal {G'}={G-M: M text { is a matching of }G}. Furthermore, we determine the value of mathrm{pr}(K_{n}, P_{l}) for sufficiently large n and the exact value of mathrm{pr}(K_{n}, G), where G is C_{5}, C_{6} and K_{4}^{-}, respectively. Also, we give an upper bound and a lower bound of mathrm{pr}(K_{n}, K_{2,3}).

Universal and unavoidable graphs

AbstractThe Turán number ex(n, H) of a graph H is the maximal number of edges in an H-free graph on n vertices. In 1983, Chung and Erdős asked which graphs H with e edges minimise ex(n, H). They resolved this question asymptotically for most of the range of e and asked to complete the picture. In this paper, we answer their question by resolving all remaining cases. Our result translates directly to the setting of universality, a well-studied notion of finding graphs which contain every graph belonging to a certain family. In this setting, we extend previous work done by Babai, Chung, Erdős, Graham and Spencer, and by Alon and Asodi.

Extremal even-cycle-free subgraphs of the complete transposition graphs

Given graphs G and H, the generalized Turán number ex(G,H) is the maximum number of edges in an H-free subgraph of G. In this paper, we obtain an asymptotic upper bound on ex(CTn,C2ℓ) for any n≥3 and ℓ≥2, where C2ℓ is the cycle of length 2ℓ and CTn is the complete transposition graph which is defined as the Cayley graph on the symmetric group Sn with respect to the set of all transpositions of Sn.

Exact bipartite Turán numbers of large even cycles

AbstractThe bipartite Turán number, denoted by , is the maximum number of edges in an ‐free bipartite graph with two parts of sizes and , respectively. In this article, we prove for any positive integers with , which confirms (in a strong form) the unsolved case of a conjecture of Győri. Let be a bipartite graph with and . Suppose that for some . We prove that contains cycles of all even lengths from 4 to if . This result sharpens a theorem on long cycles in bipartite graphs due to Jackson. As a main tool, for a longest cycle in a bipartite graph, we prove an upper bound of the number of edges which are incident with at most one vertex in , which is a bipartite analogue of a classical theorem of Bondy. Our proof technique is structural.

On even-cycle-free subgraphs of the doubled Johnson graphs

The generalized Turán number ex(G,H) is the maximum number of edges in an H-free subgraph of a graph G. It is an important extension of the classical Turán number ex(n,H), which is the maximum number of edges in a graph with n vertices that does not contain H as a subgraph. In this paper, we consider the maximum number of edges in an even-cycle-free subgraph of the doubled Johnson graphs J(n;k,k+1), which are bipartite subgraphs of hypercube graphs. We give an upper bound for ex(J(n;k,k+1),C2r) with any fixed k∈Z+ and any n∈Z+ with n≥2k+1. We also give an upper bound for ex(J(2k+1;k,k+1),C2r) with any k∈Z+, where J(2k+1;k,k+1) is known as doubled Odd graph O˜k+1. This bound induces that the number of edges in any C2r-free subgraph of O˜k+1 is o(e(O˜k+1)) for r≥6, which also implies a Ramsey-type result.

The Lower and Upper Bounds of Turán Number for Odd Wheels

The Turan number for a graph H, denoted by $$\text {ex}(n,H)$$ , is the maximum number of edges in any simple graph with n vertices which doesn’t contain H as a subgraph. In this paper we find the lower and upper bounds for $$\text { ex}(n,W_{2t+1})$$ . We show that if $$n\ge 4t$$ , then $$\text { ex}(n,W_{2t+1})\ge \left\lfloor \lfloor \frac{2n+t}{4}\rfloor (n+\frac{t-1}{2}-\lfloor \frac{2n+t}{4}\rfloor )\right\rfloor +1.$$ We also show that for sufficiently large n and $$t\ge 5$$ , $$\text { ex}(n,W_{2t+1})\le \frac{ n^2 }{4}+{t-1\over 2}n$$ . Moreover we find the exact value of the Turan number for $$W_9$$ . That is, we show that for sufficiently large n, $$\text { ex}(n,W_9)= \lfloor \frac{n^2}{4}\rfloor +\lceil \frac{3}{4}n\rceil +1$$ .

Lagrangian densities of short 3-uniform linear paths and Turán numbers of their extensions

For a fixed positive integer n and an r-uniform hypergraph H, the Turan number ex(n, H) is the maximum number of edges in an H-free r-uniform hypergraph on n vertices, and the Lagrangian density of H is defined as $$\pi _{\lambda }(H)=\sup \{r! \lambda (G) : G \;\text {is an}\; H\text {-free} \;r\text {-uniform hypergraph}\}$$ , where $$\lambda (G)=\max \{\sum _{e \in G}\prod \limits _{i\in e}x_{i}: x_i\ge 0 \;\text {and}\; \sum _{i\in V(G)} x_i=1\}$$ is the Lagrangian of G. For an r-uniform hypergraph H on t vertices, it is clear that $$\pi _{\lambda }(H)\ge r!\lambda {(K_{t-1}^r)}$$ . Let us say that an r-uniform hypergraph H on t vertices is $$\lambda $$ -perfect if $$\pi _{\lambda }(H)= r!\lambda {(K_{t-1}^r)}$$ . A result of Motzkin and Straus implies that all graphs are $$\lambda $$ -perfect. It is interesting to explore what kind of hypergraphs are $$\lambda $$ -perfect. A conjecture in [22] proposes that every sufficiently large r-uniform linear hypergraph is $$\lambda $$ -perfect. In this paper, we investigate whether the conjecture holds for linear 3-uniform paths. Let $$P_t=\{e_1, e_2, \dots , e_t\}$$ be the linear 3-uniform path of length t, that is, $$|e_i|=3$$ , $$|e_i \cap e_{i+1}|=1$$ and $$e_i \cap e_j=\emptyset $$ if $$|i-j|\ge 2$$ . We show that $$P_3$$ and $$P_4$$ are $$\lambda $$ -perfect. Applying the results on Lagrangian densities, we determine the Turan numbers of their extensions.

On the rational Turán exponents conjecture

The extremal number ex(n,F) of a graph F is the maximum number of edges in an n-vertex graph not containing F as a subgraph. A real number r∈[1,2] is realisable if there exists a graph F with ex(n,F)=Θ(nr). Several decades ago, Erdős and Simonovits conjectured that every rational number in [1,2] is realisable. Despite decades of effort, the only known realisable numbers are 0,1,75,2, and the numbers of the form 1+1m, 2−1m, 2−2m for integers m≥1. In particular, it is not even known whether the set of all realisable numbers contains a single limit point other than the two numbers 1 and 2.In this paper, we make progress on the conjecture of Erdős and Simonovits. First, we show that 2−ab is realisable for any integers a,b≥1 with b>a and b≡±1(moda). This includes all previously known ones, and gives infinitely many limit points 2−1m in the set of all realisable numbers as a consequence.Secondly, we propose a conjecture on subdivisions of bipartite graphs. Apart from being interesting on its own, we show that, somewhat surprisingly, this subdivision conjecture in fact implies that every rational number between 1 and 2 is realisable.