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 $k=0$.

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 On Perfect Packings in Dense Graphs

10.37236/3173 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
József Balogh ◽  
Alexandr Kostochka ◽  
Andrew Treglown
Keyword(s):  
Minimum Degree ◽  
Degree Sequence ◽  
Edge Density ◽  
Structural Result ◽  
Dense Graphs ◽  
Delta G ◽  
Vertex Disjoint ◽  
Density Threshold

We say that a graph $G$ has a perfect $H$-packing if there exists a set of vertex-disjoint copies of $H$ which cover all the vertices in $G$. We consider various problems concerning perfect $H$-packings: Given $n, r , D \in \mathbb N$, we characterise the edge density threshold that ensures a perfect $K_r$-packing in any graph $G$ on $n$ vertices and with minimum degree $\delta (G) \geq D$. We also give two conjectures concerning degree sequence conditions which force a graph to contain a perfect $H$-packing. Other related embedding problems are also considered. Indeed, we give a structural result concerning $K_r$-free graphs that satisfy a certain degree sequence condition.

scholarly journals Cover time of a random graph with given degree sequence

2010 ◽  
Vol DMTCS Proceedings vol. AM,... (Proceedings) ◽  
Author(s):  
Mohammed Abdullah ◽  
Colin Cooper ◽  
Alan Frieze
Keyword(s):  
Random Graph ◽  
High Probability ◽  
Minimum Degree ◽  
Degree Sequence ◽  
Average Degree ◽  
Cover Time ◽  
Vertex Set ◽  
International Audience

International audience In this paper we establish the cover time of a random graph $G(\textbf{d})$ chosen uniformly at random from the set of graphs with vertex set $[n]$ and degree sequence $\textbf{d}$. We show that under certain restrictions on $\textbf{d}$, the cover time of $G(\textbf{d})$ is with high probability asymptotic to $\frac{d-1}{ d-2} \frac{\theta}{ d}n \log n$. Here $\theta$ is the average degree and $d$ is the $\textit{effective minimum degree}$. The effective minimum degree is the first entry in the sorted degree sequence which occurs order $n$ times.

scholarly journals Counting Trees in Graphs

10.37236/6084 ◽  
2016 ◽  
Vol 23 (3) ◽  
Author(s):  
Jacques Verstraete ◽  
Dhruv Mubayi
Keyword(s):  
Lower Bound ◽  
Convex Function ◽  
Disjoint Union ◽  
Minimum Degree ◽  
Degree Sequence ◽  
Average Degree ◽  
Vertex Graph ◽  
Counting Trees

Erdős and Simonovits proved that the number of paths of length $t$ in an $n$-vertex graph of average degree $d$ is at least $(1 - \delta) nd(d - 1) \cdots (d - t + 1)$, where $\delta = (\log d)^{-1/2 + o(1)}$ as $d \rightarrow \infty$. In this paper, we strengthen and generalize this result as follows. Let $T$ be a tree with $t$ edges. We prove that for any $n$-vertex graph $G$ of average degree $d$ and minimum degree greater than $t$, the number of labelled copies of $T$ in $G$ is at least \[(1 - \varepsilon) n d(d - 1) \cdots (d - t + 1)\] where $\varepsilon = O(d^{-2})$ as $d \rightarrow \infty$. This bound is tight except for the term $1 - \varepsilon$, as shown by a disjoint union of cliques. Our proof is obtained by first showing a lower bound that is a convex function of the degree sequence of $G$, and this answers a question of Dellamonica et. al.

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 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 Cliques in Graphs With Bounded Minimum Degree

2012 ◽  
Vol 21 (3) ◽  
pp. 457-482 ◽  
Author(s):  
ALLAN LO
Keyword(s):  
Minimum Degree ◽  
Extremal Graphs ◽  
Minimum Number

Let kr(n, δ) be the minimum number of r-cliques in graphs with n vertices and minimum degree at least δ. We evaluate kr(n, δ) for δ ≤ 4n/5 and some other cases. Moreover, we give a construction which we conjecture to give all extremal graphs (subject to certain conditions on n, δ and r).

On the double total dominator chromatic number of graphs

2021 ◽  
pp. 2150154
Author(s):  
Fairouz Beggas ◽  
Hamamache Kheddouci ◽  
Walid Marweni
Keyword(s):  
Chromatic Number ◽  
Minimum Degree ◽  
Domination Number ◽  
Vertex Coloring ◽  
Total Domination Number ◽  
Close Relationship ◽  
Minimum Number ◽  
Number Formula ◽  
Dominator Coloring ◽  
Proper Vertex

In this paper, we introduce and study a new coloring problem of graphs called the double total dominator coloring. A double total dominator coloring of a graph [Formula: see text] with minimum degree at least 2 is a proper vertex coloring of [Formula: see text] such that each vertex has to dominate at least two color classes. The minimum number of colors among all double total dominator coloring of [Formula: see text] is called the double total dominator chromatic number, denoted by [Formula: see text]. Therefore, we establish the close relationship between the double total dominator chromatic number [Formula: see text] and the double total domination number [Formula: see text]. We prove the NP-completeness of the problem. We also examine the effects on [Formula: see text] when [Formula: see text] is modified by some operations. Finally, we discuss the [Formula: see text] number of square of trees by giving some bounds.

Sign in / Sign up

Export Citation Format

Share Document