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 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 The Random Graph Intuition for the Tournament Game

2015 ◽  
Vol 25 (1) ◽  
pp. 76-88 ◽  
Author(s):  
DENNIS CLEMENS ◽  
HEIDI GEBAUER ◽  
ANITA LIEBENAU
Keyword(s):  
Lower Bound ◽  
Random Graph ◽  
Complete Graph ◽  
Upper Bound ◽  
Winning Strategy ◽  
Constant Factor ◽  
Additive Constant ◽  
Underlying Graph ◽  
Straightforward Application

In the tournament game two players, called Maker and Breaker, alternately take turns in claiming an unclaimed edge of the complete graph Kn and selecting one of the two possible orientations. Before the game starts, Breaker fixes an arbitrary tournament Tk on k vertices. Maker wins if, at the end of the game, her digraph contains a copy of Tk; otherwise Breaker wins. In our main result, we show that Maker has a winning strategy for k = (2 − o(1))log2n, improving the constant factor in previous results of Beck and the second author. This is asymptotically tight since it is known that for k = (2 − o(1))log2n Breaker can prevent the underlying graph of Maker's digraph from containing a k-clique. Moreover, the precise value of our lower bound differs from the upper bound only by an additive constant of 12.We also discuss the question of whether the random graph intuition, which suggests that the threshold for k is asymptotically the same for the game played by two ‘clever’ players and the game played by two ‘random’ players, is supported by the tournament game. It will turn out that, while a straightforward application of this intuition fails, a more subtle version of it is still valid.Finally, we consider the orientation game version of the tournament game, where Maker wins the game if the final digraph – also containing the edges directed by Breaker – possesses a copy of Tk. We prove that in that game Breaker has a winning strategy for k = (4 + o(1))log2n.

scholarly journals Improved bounds for the Erdős-Rogers function

2020 ◽  
Author(s):  
Timothy Gowers ◽  
Oliver Janzer
Keyword(s):  
Graph Theory ◽  
Lower Bound ◽  
Complete Graph ◽  
Related Problem ◽  
Independent Set ◽  
Large Structure ◽  
Induced Subgraphs ◽  
Free Graph ◽  
Induced Subgraph ◽  
Complete Subgraph

[Ramsey's Theorem](https://en.wikipedia.org/wiki/Ramsey%27s_theorem) is one of the most prominent results in graph theory. In its simplest form, it asserts that every sufficiently large two-edge-colored complete graph contains a large monochromatic complete subgraph. This theorem has been generalized to a plethora of statements asserting that every sufficiently large structure of a given kind contains a large "tame" substructure. The article concerns a closely related problem: for a structure with a given property, find a substructure possessing an even stronger property. For example, what is the largest $K_3$-free induced subgraph of an $n$-vertex $K_4$-free graph? The answer to this question is approximately $n^{1/2}$. The lower bound is easy. If a given graph has a vertex of degree at least $n^{1/2}$, then its neighbors induce a $K_3$-free subgraph with at least $n^{1/2}$ vertices. Otherwise, a greedy procedure yields an independent set of size almost $n^{1/2}$. The argument generalizes to $K_s$-free induced subgraphs of $K_{s+1}$-free graphs. Dudek, Retter and Rödl provided a construction showing that the exponent $1/2$ cannot be improved and asked whether the same is the case for $K_s$-free induced subgraphs of $K_{s+2}$-free graphs. The authors answer this question by providing a construction of $K_{s+2}$-free $n$-vertex graphs with no $K_s$-free induced subgraph with $n^{\alpha_s}$ vertices with $\alpha_s<1/2$ for every $s\ge 3$. Their arguments extend to the case of $K_t$-free graphs with no large $K_s$-free induced subgraph for $s+2\le t\le 2s-1$ and $s\ge 3$.

scholarly journals Structure and Colour in Triangle-Free Graphs

10.37236/9267 ◽  
2021 ◽  
Vol 28 (2) ◽  
Author(s):  
N. R. Aravind ◽  
Stijn Cambie ◽  
Wouter Cames van Batenburg ◽  
Rémi De Joannis de Verclos ◽  
Ross J. Kang ◽  
...  
Keyword(s):  
Graph Theory ◽  
Chromatic Number ◽  
Short Proof ◽  
Independent Set ◽  
Extremal Graph ◽  
Extremal Graph Theory ◽  
Constant Multiple ◽  
Free Graph ◽  
Related Notion ◽  
Induced Path

Motivated by a recent conjecture of the first author, we prove that every properly coloured triangle-free graph of chromatic number $\chi$ contains a rainbow independent set of size $\lceil\frac12\chi\rceil$. This is sharp up to a factor $2$. This result and its short proof have implications for the related notion of chromatic discrepancy. Drawing inspiration from both structural and extremal graph theory, we conjecture that every triangle-free graph of chromatic number $\chi$ contains an induced cycle of length $\Omega(\chi\log\chi)$ as $\chi\to\infty$. Even if one only demands an induced path of length $\Omega(\chi\log\chi)$, the conclusion would be sharp up to a constant multiple. We prove it for regular girth $5$ graphs and for girth $21$ graphs. As a common strengthening of the induced paths form of this conjecture and of Johansson's theorem (1996), we posit the existence of some $c >0$ such that for every forest $H$ on $D$ vertices, every triangle-free and induced $H$-free graph has chromatic number at most $c D/\log D$. We prove this assertion with 'triangle-free' replaced by 'regular girth 5'.

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 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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