scholarly journals Dense $H$-Free Graphs are Almost $(\chi(H)-1)$-Partite

10.37236/293 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Peter Allen
Keyword(s):  
Extremal Function ◽  
Minimum Degree ◽  
Regularity Lemma ◽  
Correct Order ◽  
Large N ◽  
Vertex Graph ◽  
Order Of Magnitude ◽  
Delta G ◽  
General Graphs

By using the Szemerédi Regularity Lemma, Alon and Sudakov recently extended the classical Andrásfai-Erdős-Sós theorem to cover general graphs. We prove, without using the Regularity Lemma, that the following stronger statement is true. Given any $(r+1)$-partite graph $H$ whose smallest part has $t$ vertices, there exists a constant $C$ such that for any given $\varepsilon>0$ and sufficiently large $n$ the following is true. Whenever $G$ is an $n$-vertex graph with minimum degree $$\delta(G)\geq\left(1-{3\over 3r-1}+\varepsilon\right)n,$$ either $G$ contains $H$, or we can delete $f(n,H)\leq Cn^{2-{1\over t}}$ edges from $G$ to obtain an $r$-partite graph. Further, we are able to determine the correct order of magnitude of $f(n,H)$ in terms of the Zarankiewicz extremal function.

scholarly journals Stability for Vertex Cycle Covers

10.37236/5185 ◽  
2017 ◽  
Vol 24 (3) ◽  
Author(s):  
József Balogh ◽  
Frank Mousset ◽  
Jozef Skokan
Keyword(s):  
Minimum Degree ◽  
Independent Set ◽  
Stability Result ◽  
Natural Generalization ◽  
Regularity Lemma ◽  
Property A ◽  
Vertex Graph ◽  
Szemeredi's Regularity Lemma ◽  
Cycle Covers ◽  
Positive Integers

In 1996 Kouider and Lonc proved the following natural generalization of Dirac's Theorem: for any integer $k\geq 2$, if $G$ is an $n$-vertex graph with minimum degree at least $n/k$, then there are $k-1$ cycles in $G$ that together cover all the vertices.This is tight in the sense that there are $n$-vertex graphs that have minimum degree $n/k-1$ and that do not contain $k-1$ cycles with this property. A concrete example is given by $I_{n,k} = K_n\setminus K_{(k-1)n/k+1}$ (an edge-maximal graph on $n$ vertices with an independent set of size $(k-1)n/k+1$). This graph has minimum degree $n/k-1$ and cannot be covered with fewer than $k$ cycles. More generally, given positive integers $k_1,\dotsc,k_r$ summing to $k$, the disjoint union $I_{k_1n/k,k_1}+ \dotsb + I_{k_rn/k,k_r}$ is an $n$-vertex graph with the same properties.In this paper, we show that there are no extremal examples that differ substantially from the ones given by this construction. More precisely, we obtain the following stability result: if a graph $G$ has $n$ vertices and minimum degree nearly $n/k$, then it either contains $k-1$ cycles covering all vertices, or else it must be close (in ‘edit distance') to a subgraph of $I_{k_1n/k,k_1}+ \dotsb + I_{k_rn/k,k_r}$, for some sequence $k_1,\dotsc,k_r$ of positive integers that sum to $k$.Our proof uses Szemerédi's Regularity Lemma and the related machinery.

Monochromatic paths and cycles in 2-edge-coloured graphs with large minimum degree

2021 ◽  
pp. 1-14
Author(s):  
József Balogh ◽  
Alexandr Kostochka ◽  
Mikhail Lavrov ◽  
Xujun Liu
Keyword(s):  
Complete Graph ◽  
Minimum Degree ◽  
Dense Subgraph ◽  
Degree Condition ◽  
Vertex Graph ◽  
Delta G ◽  
Edge Colouring ◽  
Monochromatic Copy

Abstract A graph G arrows a graph H if in every 2-edge-colouring of G there exists a monochromatic copy of H. Schelp had the idea that if the complete graph $K_n$ arrows a small graph H, then every ‘dense’ subgraph of $K_n$ also arrows H, and he outlined some problems in this direction. Our main result is in this spirit. We prove that for every sufficiently large n, if $n = 3t+r$ where $r \in \{0,1,2\}$ and G is an n-vertex graph with $\delta(G) \ge (3n-1)/4$ , then for every 2-edge-colouring of G, either there are cycles of every length $\{3, 4, 5, \dots, 2t+r\}$ of the same colour, or there are cycles of every even length $\{4, 6, 8, \dots, 2t+2\}$ of the samecolour. Our result is tight in the sense that no longer cycles (of length $>2t+r$ ) can be guaranteed and the minimum degree condition cannot be reduced. It also implies the conjecture of Schelp that for every sufficiently large n, every $(3t-1)$ -vertex graph G with minimum degree larger than $3|V(G)|/4$ arrows the path $P_{2n}$ with 2n vertices. Moreover, it implies for sufficiently large n the conjecture by Benevides, Łuczak, Scott, Skokan and White that for $n=3t+r$ where $r \in \{0,1,2\}$ and every n-vertex graph G with $\delta(G) \ge 3n/4$ , in each 2-edge-colouring of G there exists a monochromatic cycle of length at least $2t+r$ .

scholarly journals The bandwidth theorem for locally dense graphs

2020 ◽  
Vol 8 ◽  
Author(s):  
Katherine Staden ◽  
Andrew Treglown
Keyword(s):  
Maximum Degree ◽  
Blow Up ◽  
Minimum Degree ◽  
Bounded Degree ◽  
Dense Graphs ◽  
Vertex Graph ◽  
Delta G

Abstract The bandwidth theorem of Böttcher, Schacht, and Taraz [Proof of the bandwidth conjecture of Bollobás andKomlós, Mathematische Annalen, 2009] gives a condition on the minimum degree of an n-vertex graph G that ensures G contains every r-chromatic graph H on n vertices of bounded degree and of bandwidth $o(n)$ , thereby proving a conjecture of Bollobás and Komlós [The Blow-up Lemma, Combinatorics, Probability, and Computing, 1999]. In this paper, we prove a version of the bandwidth theorem for locally dense graphs. Indeed, we prove that every locally dense n-vertex graph G with $\delta (G)> (1/2+o(1))n$ contains as a subgraph any given (spanning) H with bounded maximum degree and sublinear bandwidth.

scholarly journals Maximum Planar Subgraphs in Dense Graphs

10.37236/3041 ◽  
2013 ◽  
Vol 20 (3) ◽  
Author(s):  
Peter Allen ◽  
Jozef Skokan ◽  
Andreas Würfl
Keyword(s):  
Minimum Degree ◽  
Large N ◽  
Dense Graphs ◽  
Vertex Graph ◽  
Planar Subgraphs

Kühn, Osthus and Taraz showed that for each $\gamma>0$ there exists $C$ such that any $n$-vertex graph with minimum degree $\gamma n$ contains a planar subgraph with at least $2n-C$ edges. We find the optimum value of $C$ for all $\gamma< 1/2$ and sufficiently large $n$.

scholarly journals Proof of the $(n/2-n/2-n/2)$ Conjecture for Large $n$

10.37236/514 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Yi Zhao
Keyword(s):  
Upper Bound ◽  
Ramsey Number ◽  
Regularity Lemma ◽  
Large N ◽  
Vertex Graph ◽  
Approximate Version

A conjecture of Loebl, also known as the $(n/2 - n/2 - n/2)$ Conjecture, states that if $G$ is an $n$-vertex graph in which at least $n/2$ of the vertices have degree at least $n/2$, then $G$ contains all trees with at most $n/2$ edges as subgraphs. Applying the Regularity Lemma, Ajtai, Komlós and Szemerédi proved an approximate version of this conjecture. We prove it exactly for sufficiently large $n$. This immediately gives a tight upper bound for the Ramsey number of trees, and partially confirms a conjecture of Burr and Erdős.

scholarly journals On degree-sequence characterization and the extremal number of edges for various Hamiltonian properties under fault tolerance

2016 ◽  
Vol Vol. 17 no. 3 (Graph Theory) ◽  
Author(s):  
Shih-Yan Chen ◽  
Shin-Shin Kao ◽  
Hsun Su
Keyword(s):  
Minimum Degree ◽  
Degree Sequence ◽  
Sequence Characterization ◽  
Hamiltonian Connected ◽  
Vertex Graph ◽  
Minimum Number ◽  
Hamiltonian Connectivity ◽  
International Audience ◽  
Delta G ◽  
Hamiltonian Properties

International audience Assume that $n, \delta ,k$ are integers with $0 \leq k < \delta < n$. Given a graph $G=(V,E)$ with $|V|=n$. The symbol $G-F, F \subseteq V$, denotes the graph with $V(G-F)=V-F$, and $E(G-F)$ obtained by $E$ after deleting the edges with at least one endvertex in $F$. $G$ is called <i>$k$-vertex fault traceable</i>, <i>$k$-vertex fault Hamiltonian</i>, or <i>$k$-vertex fault Hamiltonian-connected</i> if $G-F$ remains traceable, Hamiltonian, and Hamiltonian-connected for all $F$ with $0 \leq |F| \leq k$, respectively. The notations $h_1(n, \delta ,k)$, $h_2(n, \delta ,k)$, and $h_3(n, \delta ,k)$ denote the minimum number of edges required to guarantee an $n$-vertex graph with minimum degree $\delta (G) \geq \delta$ to be $k$-vertex fault traceable, $k$-vertex fault Hamiltonian, and $k$-vertex fault Hamiltonian-connected, respectively. In this paper, we establish a theorem which uses the degree sequence of a given graph to characterize the $k$-vertex fault traceability/hamiltonicity/Hamiltonian-connectivity, respectively. Then we use this theorem to obtain the formulas for $h_i(n, \delta ,k)$ for $1 \leq i \leq 3$, which improves and extends the known results for $k=0$.

scholarly journals Sharp Minimum Degree Conditions for the Existence of Disjoint Theta Graphs

10.37236/9670 ◽  
2021 ◽  
Vol 28 (3) ◽  
Author(s):  
Emily Marshall ◽  
Michael Santana
Keyword(s):  
Minimum Degree ◽  
Degree Condition ◽  
Sharp Minimum ◽  
Disjoint Cycles ◽  
Vertex Graph ◽  
Degree Conditions ◽  
Delta G

In 1963, Corrádi and Hajnal showed that if $G$ is an $n$-vertex graph with  $n \ge 3k$ and $\delta(G) \ge 2k$, then $G$ will contain $k$ disjoint cycles; furthermore, this result is best possible, both in terms of the number of vertices as well as the minimum degree. In this paper we focus on an analogue of this result for theta graphs.  Results from Kawarabayashi and Chiba et al. showed that if $n = 4k$ and $\delta(G) \ge \lceil \frac{5}{2}k \rceil$, or if $n$ is large with respect to $k$ and $\delta(G) \ge 2k+1$, respectively, then $G$ contains $k$ disjoint theta graphs.  While the minimum degree condition in both results are sharp for the number of vertices considered, this leaves a gap in which no sufficient minimum degree condition is known. Our main result in this paper resolves this by showing if $n \ge 4k$ and $\delta(G) \ge \lceil \frac{5}{2}k\rceil$, then $G$ contains $k$ disjoint theta graphs. Furthermore, we show this minimum degree condition is sharp for more than just $n = 4k$, and we discuss how and when the sharp minimum degree condition may transition from $\lceil \frac{5}{2}k\rceil$ to $2k+1$.

scholarly journals The Bandwidth Theorem in sparse graphs

2020 ◽  
Author(s):  
Peter Allen ◽  
Julia Böttcher ◽  
Julia Ehrenmüller ◽  
Anusch Taraz
Keyword(s):  
Hamiltonian Cycle ◽  
Blow Up ◽  
Minimum Degree ◽  
Main Tool ◽  
Sparse Graphs ◽  
Spanning Subgraphs ◽  
Dense Graphs ◽  
Vertex Graph ◽  
First Results ◽  
Delta G

One of the first results in graph theory was Dirac's theorem which claims that if the minimum degree in a graph is at least half of the number of vertices, then it contains a Hamiltonian cycle. This result has inspired countless other results all stating that in dense graphs we can find sparse spanning subgraphs. Along these lines, one of the most far-reaching results is the celebrated _Bandwidth Theorem_, proved around 10 years ago by Böttcher, Schacht, and Taraz. It states, rougly speaking, that every $n$-vertex graph with minimum degree at least $\left( \frac{r-1}{r} + o(1)\right) n$ contains a copy of all $n$-vertex graphs $H$ such that $\chi(H) \leq r$, $\Delta (H) = O(1)$, and the bandwidth of $H$ is $o(n)$. This was conjectured earlier by Bollobás and Komlós. The proof is using the Regularity method based on the Regularity Lemma and the Blow-up Lemma. Ever since the Bandwith Theorem came out, it has been open whether one could prove a similar statement for sparse random graphs. In this remarkable, deep paper the authors do just that, they establish sparse random analogues of the Bandwidth Theorem. In particular, the authors show that, for every positive integer $\Delta$, if $p \gg \left(\frac{\log{n}}{n}\right)^{1/\Delta}$, then asymptotically almost surely, every subgraph $G\subseteq G(n, p)$ with $\delta(G) \geq \left( \frac{r-1}{r} + o(1)\right) np$ contains a copy of every $r$-colourable spanning (i.e., $n$-vertex) graph $H$ with maximum degree at most $\Delta$ and bandwidth $o(n)$, provided that $H$ contains at least $C p^{-2}$ vertices that do not lie on a triangle (of $H$). (The requirement about vertices not lying on triangles is necessary, as pointed out by Huang, Lee, and Sudakov.) The main tool used in the proof is the recent monumental sparse Blow-up Lemma due to Allen, Böttcher, Hàn, Kohayakawa, and Person.

scholarly journals Hereditary quasirandomness without regularity

2017 ◽  
Vol 164 (3) ◽  
pp. 385-399 ◽  
Author(s):  
DAVID CONLON ◽  
JACOB FOX ◽  
BENNY SUDAKOV
Keyword(s):  
Alternative Proof ◽  
Regularity Lemma ◽  
Original Proof ◽  
Vertex Graph

AbstractA result of Simonovits and Sós states that for any fixed graph H and any ε > 0 there exists δ > 0 such that if G is an n-vertex graph with the property that every S ⊆ V(G) contains pe(H) |S|v(H) ± δ nv(H) labelled copies of H, then G is quasirandom in the sense that every S ⊆ V(G) contains $\frac{1}{2}$p|S|2± ε n2 edges. The original proof of this result makes heavy use of the regularity lemma, resulting in a bound on δ−1 which is a tower of twos of height polynomial in ε−1. We give an alternative proof of this theorem which avoids the regularity lemma and shows that δ may be taken to be linear in ε when H is a clique and polynomial in ε for general H. This answers a problem raised by Simonovits and Sós.

scholarly journals A New Bound on the Domination Number of Graphs with Minimum Degree Two

10.37236/499 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Michael A. Henning ◽  
Ingo Schiermeyer ◽  
Anders Yeo
Keyword(s):  
Connected Graph ◽  
Minimum Degree ◽  
Domination Number ◽  
Degree Condition ◽  
Cut Vertex ◽  
Delta G ◽  
Vertex Disjoint ◽  
Special Cycle

For a graph $G$, let $\gamma(G)$ denote the domination number of $G$ and let $\delta(G)$ denote the minimum degree among the vertices of $G$. A vertex $x$ is called a bad-cut-vertex of $G$ if $G-x$ contains a component, $C_x$, which is an induced $4$-cycle and $x$ is adjacent to at least one but at most three vertices on $C_x$. A cycle $C$ is called a special-cycle if $C$ is a $5$-cycle in $G$ such that if $u$ and $v$ are consecutive vertices on $C$, then at least one of $u$ and $v$ has degree $2$ in $G$. We let ${\rm bc}(G)$ denote the number of bad-cut-vertices in $G$, and ${\rm sc}(G)$ the maximum number of vertex disjoint special-cycles in $G$ that contain no bad-cut-vertices. We say that a graph is $(C_4,C_5)$-free if it has no induced $4$-cycle or $5$-cycle. Bruce Reed [Paths, stars and the number three. Combin. Probab. Comput. 5 (1996), 277–295] showed that if $G$ is a graph of order $n$ with $\delta(G) \ge 3$, then $\gamma(G) \le 3n/8$. In this paper, we relax the minimum degree condition from three to two. Let $G$ be a connected graph of order $n \ge 14$ with $\delta(G) \ge 2$. As an application of Reed's result, we show that $\gamma(G) \le \frac{1}{8} ( 3n + {\rm sc}(G) + {\rm bc}(G))$. As a consequence of this result, we have that (i) $\gamma(G) \le 2n/5$; (ii) if $G$ contains no special-cycle and no bad-cut-vertex, then $\gamma(G) \le 3n/8$; (iii) if $G$ is $(C_4,C_5)$-free, then $\gamma(G) \le 3n/8$; (iv) if $G$ is $2$-connected and $d_G(u) + d_G(v) \ge 5$ for every two adjacent vertices $u$ and $v$, then $\gamma(G) \le 3n/8$. All bounds are sharp.

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