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 On size, radius and minimum degree

2014 ◽  
Vol Vol. 16 no. 1 (Graph Theory) ◽  
Author(s):  
Simon Mukwembi
Keyword(s):  
Graph Theory ◽  
Upper Bound ◽  
Connected Graph ◽  
Minimum Degree ◽  
Classical Theorem ◽  
International Audience

Graph Theory International audience Let G be a finite connected graph. We give an asymptotically tight upper bound on the size of G in terms of order, radius and minimum degree. Our result is a strengthening of an old classical theorem of Vizing (1967) if minimum degree is prescribed.

scholarly journals Extremal Graph Theory for Metric Dimension and Diameter

10.37236/302 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Carmen Hernando ◽  
Mercè Mora ◽  
Ignacio M. Pelayo ◽  
Carlos Seara ◽  
David R. Wood
Keyword(s):  
Graph Theory ◽  
Upper Bound ◽  
Connected Graph ◽  
Extremal Graph ◽  
Extremal Graph Theory ◽  
Metric Dimension ◽  
Resolving Set ◽  
Maximum Order ◽  
Minimum Order ◽  
Minimum Cardinality

A set of vertices $S$ resolves a connected graph $G$ if every vertex is uniquely determined by its vector of distances to the vertices in $S$. The metric dimension of $G$ is the minimum cardinality of a resolving set of $G$. Let ${\cal G}_{\beta,D}$ be the set of graphs with metric dimension $\beta$ and diameter $D$. It is well-known that the minimum order of a graph in ${\cal G}_{\beta,D}$ is exactly $\beta+D$. The first contribution of this paper is to characterise the graphs in ${\cal G}_{\beta,D}$ with order $\beta+D$ for all values of $\beta$ and $D$. Such a characterisation was previously only known for $D\leq2$ or $\beta\leq1$. The second contribution is to determine the maximum order of a graph in ${\cal G}_{\beta,D}$ for all values of $D$ and $\beta$. Only a weak upper bound was previously known.

scholarly journals Triangle-Free Subgraphs of Random Graphs

2017 ◽  
Vol 27 (2) ◽  
pp. 141-161
Author(s):  
PETER ALLEN ◽  
JULIA BÖTTCHER ◽  
YOSHIHARU KOHAYAKAWA ◽  
BARNABY ROBERTS
Keyword(s):  
Graph Theory ◽  
Random Graph ◽  
Random Graphs ◽  
Minimum Degree ◽  
Discrete Math ◽  
Constant Factor ◽  
Extremal Graph ◽  
Extremal Graph Theory ◽  
Spanning Subgraph

Recently there has been much interest in studying random graph analogues of well-known classical results in extremal graph theory. Here we follow this trend and investigate the structure of triangle-free subgraphs of G(n, p) with high minimum degree. We prove that asymptotically almost surely each triangle-free spanning subgraph of G(n, p) with minimum degree at least (2/5 + o(1))pn is (p−1n)-close to bipartite, and each spanning triangle-free subgraph of G(n, p) with minimum degree at least (1/3 + ϵ)pn is O(p−1n)-close to r-partite for some r = r(ϵ). These are random graph analogues of a result by Andrásfai, Erdős and Sós (Discrete Math.8 (1974), 205–218), and a result by Thomassen (Combinatorica22 (2002), 591–596). We also show that our results are best possible up to a constant factor.

scholarly journals DEGREE DISTANCE AND MINIMUM DEGREE

2012 ◽  
Vol 87 (2) ◽  
pp. 255-271 ◽  
Author(s):  
S. MUKWEMBI ◽  
S. MUNYIRA
Keyword(s):  
Upper Bound ◽  
Connected Graph ◽  
Minimum Degree ◽  
Degree Of Vertex ◽  
Degree Distance ◽  
Image Position ◽  
Appl Math ◽  
Extremal Properties

AbstractLet G be a finite connected graph of order n, minimum degree δ and diameter d. The degree distance D′(G) of G is defined as ∑ {u,v}⊆V (G)(deg u+deg v) d(u,v), where deg w is the degree of vertex w and d(u,v) denotes the distance between u and v. In this paper, we find an asymptotically sharp upper bound on the degree distance in terms of order, minimum degree and diameter. In particular, we prove that \[ D^\prime (G)\le \frac {1}{4}\,dn\biggl (n-\frac {d}{3}(\delta +1)\biggr )^2+O(n^3). \] As a corollary, we obtain the bound D′ (G)≤n4 /(9(δ+1) )+O(n3) for a graph G of order n and minimum degree δ. This result, apart from improving on a result of Dankelmann et al. [‘On the degree distance of a graph’, Discrete Appl. Math.157 (2009), 2773–2777] for graphs of given order and minimum degree, completely settles a conjecture of Tomescu [‘Some extremal properties of the degree distance of a graph’, Discrete Appl. Math.98(1999), 159–163].

scholarly journals Wiener index in graphs with given minimum degree and maximum degree

2021 ◽  
Vol vol. 23 no. 1 (Graph Theory) ◽  
Author(s):  
Peter Dankelmann ◽  
Alex Alochukwu
Keyword(s):  
Graph Theory ◽  
Upper Bound ◽  
Connected Graph ◽  
Maximum Degree ◽  
Minimum Degree ◽  
Large Degree ◽  
Wiener Index ◽  
Sharp Bound

Let $G$ be a connected graph of order $n$.The Wiener index $W(G)$ of $G$ is the sum of the distances between all unordered pairs of vertices of $G$. In this paper we show that the well-known upper bound $\big( \frac{n}{\delta+1}+2\big) {n \choose 2}$ on the Wiener index of a graph of order $n$ and minimum degree $\delta$ [M. Kouider, P. Winkler, Mean distance and minimum degree. J. Graph Theory 25 no. 1 (1997)] can be improved significantly if the graph contains also a vertex of large degree. Specifically, we give the asymptotically sharp bound $W(G) \leq {n-\Delta+\delta \choose 2} \frac{n+2\Delta}{\delta+1}+ 2n(n-1)$ on the Wiener index of a graph $G$ of order $n$, minimum degree $\delta$ and maximum degree $\Delta$. We prove a similar result for triangle-free graphs, and we determine a bound on the Wiener index of $C_4$-free graphs of given order, minimum and maximum degree and show that it is, in some sense, best possible.

scholarly journals On Winning Fast in Avoider-Enforcer Games

10.37236/328 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
János Barát ◽  
Miloš Stojaković
Keyword(s):  
Graph Theory ◽  
Lower Bound ◽  
Complete Graph ◽  
Upper Bound ◽  
Extremal Graph ◽  
Extremal Graph Theory ◽  
Additive Constant ◽  
Main Interest ◽  
Free Graph ◽  
Positional Games

We analyze the duration of the unbiased Avoider-Enforcer game for three basic positional games. All the games are played on the edges of the complete graph on $n$ vertices, and Avoider's goal is to keep his graph outerplanar, diamond-free and $k$-degenerate, respectively. It is clear that all three games are Enforcer's wins, and our main interest lies in determining the largest number of moves Avoider can play before losing. Extremal graph theory offers a general upper bound for the number of Avoider's moves. As it turns out, for all three games we manage to obtain a lower bound that is just an additive constant away from that upper bound. In particular, we exhibit a strategy for Avoider to keep his graph outerplanar for at least $2n-8$ moves, being just $6$ short of the maximum possible. A diamond-free graph can have at most $d(n)=\lceil\frac{3n-4}{2}\rceil$ edges, and we prove that Avoider can play for at least $d(n)-3$ moves. Finally, if $k$ is small compared to $n$, we show that Avoider can keep his graph $k$-degenerate for as many as $e(n)$ moves, where $e(n)$ is the maximum number of edges a $k$-degenerate graph can have.

scholarly journals Extremal graph theory and finite forcibility

2017 ◽  
Vol 61 ◽  
pp. 541-547
Author(s):  
Andrzej Grzesik ◽  
Daniel Král' ◽  
László Miklós Lovász
Keyword(s):  
Graph Theory ◽  
Extremal Graph ◽  
Extremal Graph Theory

Extremal Graph Theory

2020 ◽  
pp. 33-56
Author(s):  
Fan Chung ◽  
Ron Graham
Keyword(s):  
Graph Theory ◽  
Extremal Graph ◽  
Extremal Graph Theory

scholarly journals The Diameter of Almost Eulerian Digraphs

10.37236/429 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Peter Dankelmann ◽  
L. Volkmann
Keyword(s):  
Graph Theory ◽  
Directed Graph ◽  
Upper Bound ◽  
Minimum Degree ◽  
Additive Constant ◽  
Undirected Graphs

Soares [J. Graph Theory 1992] showed that the well known upper bound $\frac{3}{\delta+1}n+O(1)$ on the diameter of undirected graphs of order $n$ and minimum degree $\delta$ also holds for digraphs, provided they are eulerian. In this paper we investigate if similar bounds can be given for digraphs that are, in some sense, close to being eulerian. In particular we show that a directed graph of order $n$ and minimum degree $\delta$ whose arc set can be partitioned into $s$ trails, where $s\leq \delta-2$, has diameter at most $3 ( \delta+1 - \frac{s}{3})^{-1}n+O(1)$. If $s$ also divides $\delta-2$, then we show the diameter to be at most $3(\delta+1 - \frac{(\delta-2)s}{3(\delta-2)+s} )^{-1}n+O(1)$. The latter bound is sharp, apart from an additive constant. As a corollary we obtain the sharp upper bound $3( \delta+1 - \frac{\delta-2}{3\delta-5})^{-1} n + O(1)$ on the diameter of digraphs that have an eulerian trail.

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