scholarly journals On the Turán Number of Forests

10.37236/3142 ◽  
2013 ◽  
Vol 20 (2) ◽  
Author(s):  
Hong Liu ◽  
Bernard Lidicky ◽  
Cory Palmer
Keyword(s):  
Arbitrary Order ◽  
Extremal Graph ◽  
Extremal Graphs ◽  
Turán Number

The Turán number of a graph $H$, $\mathrm{ex}(n,H)$, is the maximum number of edges in a graph on $n$ vertices which does not have $H$ as a subgraph. We determine the Turán number and find the unique extremal graph for forests consisting of paths when $n$ is sufficiently large. This generalizes a result of Bushaw and Kettle [Combinatorics, Probability and Computing 20:837--853, 2011]. We also determine the Turán number and extremal graphs for forests consisting of stars of arbitrary order.

Stability and Turán Numbers of a Class of Hypergraphs via Lagrangians

2017 ◽  
Vol 26 (3) ◽  
pp. 367-405 ◽  
Author(s):  
AXEL BRANDT ◽  
DAVID IRWIN ◽  
TAO JIANG
Keyword(s):  
Positive Integer ◽  
Structural Stability ◽  
Pairwise Disjoint ◽  
Large Family ◽  
Extremal Graphs ◽  
Uniform Hypergraphs ◽  
Turán Number

Given a family ofr-uniform hypergraphs${\cal F}$(orr-graphs for brevity), the Turán number ex(n,${\cal F})$of${\cal F}$is the maximum number of edges in anr-graph onnvertices that does not contain any member of${\cal F}$. A pair {u,v} iscoveredin a hypergraphGif some edge ofGcontains {u, v}. Given anr-graphFand a positive integerp⩾n(F), wheren(F) denotes the number of vertices inF, letHFpdenote ther-graph obtained as follows. Label the vertices ofFasv1,. . .,vn(F). Add new verticesvn(F)+1,. . .,vp. For each pair of verticesvi, vjnot covered inF, add a setBi,jofr− 2 new vertices and the edge {vi, vj} ∪Bi,j, where theBi,jare pairwise disjoint over all such pairs {i, j}. We callHFpthe expanded p-clique with an embedded F. For a relatively large family ofF, we show that for all sufficiently largen, ex(n,HFp) = |Tr(n, p− 1)|, whereTr(n, p− 1) is the balanced complete (p− 1)-partiter-graph onnvertices. We also establish structural stability of near-extremal graphs. Our results generalize or strengthen several earlier results and provide a class of hypergraphs for which the Turán number is exactly determined (for largen).

scholarly journals Extremal Numbers for Odd Cycles

2014 ◽  
Vol 24 (4) ◽  
pp. 641-645 ◽  
Author(s):  
ZOLTAN FÜREDI ◽  
DAVID S. GUNDERSON
Keyword(s):  
Extremal Graph ◽  
Classical Theorem ◽  
Extremal Graphs ◽  
Complete Determination ◽  
Odd Cycles ◽  
Turán Graph

We describe theC2k+1-free graphs onnvertices with maximum number of edges. The extremal graphs are unique forn∉ {3k− 1, 3k, 4k− 2, 4k− 1}. The value ofex(n,C2k+1) can be read out from the works of Bondy [3], Woodall [14], and Bollobás [1], but here we give a new streamlined proof. The complete determination of the extremal graphs is also new.We obtain that the bound forn0(C2k+1) is 4kin the classical theorem of Simonovits, from which the unique extremal graph is the bipartite Turán graph.

scholarly journals The Turán number of the graph 3P5

Filomat ◽  
2020 ◽  
Vol 34 (10) ◽  
pp. 3395-3410
Author(s):  
Liquan Feng ◽  
Yumei Hu
Keyword(s):  
Extremal Graphs ◽  
Turán Number ◽  
Positive Integers

The Tur?n number ex(n,H) of a graph H, is the maximum number of edges in a graph of order n which does not contain H as a subgraph. Let Ex(n,H) denote all H-free graphs on n vertices with ex(n,H) edges. Let Pi denote a path consisting of i vertices, and mPi denote m disjoint copies of Pi. In this paper, we give the Tur?n number ex(n,3P5) for all positive integers n, which partly solve the conjecture proposed by L. Yuan and X. Zhang [7]. Moreover, we characterize all extremal graphs of 3P5 denoted by Ex(n, 3P5).

scholarly journals Extensions of the Erdős–Gallai theorem and Luo’s theorem

2019 ◽  
Vol 29 (1) ◽  
pp. 128-136 ◽  
Author(s):  
Bo Ning ◽  
Xing Peng
Keyword(s):  
Graph Theory ◽  
Upper Bound ◽  
Connected Graph ◽  
Minimum Degree ◽  
Extremal Graph ◽  
Extremal Graph Theory ◽  
Classical Theorem ◽  
Vertex Graph ◽  
Turán Number ◽  
Image Position

AbstractThe famous Erdős–Gallai theorem on the Turán number of paths states that every graph with n vertices and m edges contains a path with at least (2m)/n edges. In this note, we first establish a simple but novel extension of the Erdős–Gallai theorem by proving that every graph G contains a path with at least $${{(s + 1){N_{s + 1}}(G)} \over {{N_s}(G)}} + s - 1$$ edges, where Nj(G) denotes the number of j-cliques in G for 1≤ j ≤ ω(G). We also construct a family of graphs which shows our extension improves the estimate given by the Erdős–Gallai theorem. Among applications, we show, for example, that the main results of [20], which are on the maximum possible number of s-cliques in an n-vertex graph without a path with ℓ vertices (and without cycles of length at least c), can be easily deduced from this extension. Indeed, to prove these results, Luo [20] generalized a classical theorem of Kopylov and established a tight upper bound on the number of s-cliques in an n-vertex 2-connected graph with circumference less than c. We prove a similar result for an n-vertex 2-connected graph with circumference less than c and large minimum degree. We conclude this paper with an application of our results to a problem from spectral extremal graph theory on consecutive lengths of cycles in graphs.

scholarly journals Extremal graph problems with symmetrical extremal graphs. Additional chromatic conditions

1974 ◽  
Vol 7 (3-4) ◽  
pp. 349-376 ◽  
Author(s):  
M. Simonovits
Keyword(s):  
Extremal Graph ◽  
Extremal Graphs ◽  
Graph Problems

scholarly journals An Extension of Turán's Theorem, Uniqueness and Stability

10.37236/4194 ◽  
2014 ◽  
Vol 21 (4) ◽  
Author(s):  
Peter Allen ◽  
Julia Böttcher ◽  
Jan Hladký ◽  
Diana Piguet
Keyword(s):  
Extremal Graph ◽  
Extremal Graphs ◽  
Fixed Set ◽  
Vertex Graph ◽  
Turan's Theorem ◽  
Turán’S Theorem ◽  
Stability Results ◽  
Turán Graph

We determine the maximum number of edges of an $n$-vertex graph $G$ with the property that none of its $r$-cliques intersects a fixed set $M\subset V(G)$.  For $(r-1)|M|\ge n$, the $(r-1)$-partite Turán graph turns out to be the unique extremal graph. For $(r-1)|M|<n$, there is a whole family of extremal graphs, which we describe explicitly. In addition we provide corresponding stability results.

scholarly journals Extremal Graph for Intersecting Odd Cycles

10.37236/5851 ◽  
2016 ◽  
Vol 23 (2) ◽  
Author(s):  
Xinmin Hou ◽  
Yu Qiu ◽  
Boyuan Liu
Keyword(s):  
Complete Graph ◽  
Extremal Graph ◽  
Extremal Graphs ◽  
Turán Theorem ◽  
Odd Cycles ◽  
Turán Graph

An extremal graph for a graph $H$ on $n$ vertices is a graph on $n$ vertices with maximum number of edges that does not contain $H$ as a subgraph. Let $T_{n,r}$ be the Turán graph, which is the complete $r$-partite graph on $n$ vertices with part sizes that differ by at most one. The well-known Turán Theorem states that $T_{n,r}$ is the only extremal graph for complete graph $K_{r+1}$. Erdős et al. (1995) determined the extremal graphs for intersecting triangles and Chen et al. (2003) determined the maximum number of edges of the extremal graphs for intersecting cliques. In this paper, we determine the extremal graphs for intersecting odd cycles.

scholarly journals On some interconnections between combinatorial optimization and extremal graph theory

2004 ◽  
Vol 14 (2) ◽  
pp. 147-154 ◽  
Author(s):  
Dragos Cvetkovic ◽  
Pierre Hansen ◽  
Vera Kovacevic-Vujcic
Keyword(s):  
Graph Theory ◽  
Combinatorial Optimization ◽  
Efficient Algorithms ◽  
Extremal Graph ◽  
Extremal Graph Theory ◽  
Extremal Graphs ◽  
Graph Invariants ◽  
Finite Set

The uniting feature of combinatorial optimization and extremal graph theory is that in both areas one should find extrema of a function defined in most cases on a finite set. While in combinatorial optimization the point is in developing efficient algorithms and heuristics for solving specified types of problems, the extremal graph theory deals with finding bounds for various graph invariants under some constraints and with constructing extremal graphs. We analyze by examples some interconnections and interactions of the two theories and propose some conclusions.

scholarly journals Homomorphisms into Loop-Threshold Graphs

10.37236/6207 ◽  
2020 ◽  
Vol 27 (2) ◽  
Author(s):  
Jonathan Cutler ◽  
Nicholas Kass
Keyword(s):  
Independent Set ◽  
Fixed Number ◽  
Extremal Graph ◽  
Extremal Graphs ◽  
Single Vertex ◽  
Exact Answer ◽  
Small Loop ◽  
Isolated Vertex ◽  
Image Graphs ◽  
Threshold Graph

Many problems in extremal graph theory correspond to questions involving homomorphisms into a fixed image graph. Recently, there has been interest in maximizing the number of homomorphisms from graphs with a fixed number of vertices and edges into small image graphs. For the image graph $H_\text{ind}$, the graph on two adjacent vertices, one of which is looped, each homomorphism from $G$ to $H_\text{ind}$ corresponds to an independent set in $G$. It follows from the Kruskal-Katona theorem that the number of homomorphisms to $H_\text{ind}$ is maximized by the lex graph, whose edges form an initial segment of the lex order.  A loop-threshold graph is a graph built recursively from a single vertex, which may be looped or unlooped, by successively adding either a looped dominating vertex or an unlooped isolated vertex at each stage. Thus, the graph $H_\text{ind}$ is a loop-threshold graph. We survey known results for maximizing the number of homomorphisms into small loop-threshold image graphs. The only extremal homomorphism problem with a loop-threshold image graph on at most three vertices not yet solved is $H_\text{ind}\cup E_1$, where extremal graphs are the union of a lex graph and an empty graph. The only question that remains is the size of the lex component of the extremal graph. While we cannot give an exact answer for every number of vertices and edges, we establish the significance of and give bounds on $\ell(m)$, the number of vertices in the lex component of the extremal graph with $m$ edges and at least $m+1$ vertices.

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