scholarly journals Random Lifts of Graphs are Highly Connected

10.37236/2783 ◽  
2013 ◽  
Vol 20 (2) ◽  
Author(s):  
Marcin Witkowski
Keyword(s):  
Random Graphs ◽  
Maximum Degree ◽  
Asymptotic Properties ◽  
New Model ◽  
Base Graph ◽  
The Core ◽  
Similar Idea ◽  
Delta G

In this note we study asymptotic properties of random lifts of graphs introduced by Amit and Linial as a new model of random graphs. Given a base graph $G$ and an integer $n$, a random lift of $G$ is obtained by replacing each vertex of $G$ by a set of $n$ vertices, and joining these sets by random matchings whenever the corresponding vertices of $G$ are adjacent. In this paper we study connectivity properties of random lifts. We show that the size of the largest topological clique in typical random lifts, with $G$ fixed and $n\rightarrow\infty$, is equal to the maximum degree of the core of $G$ plus one. A similar idea can be used to prove that for any graph $G$ with $\delta(G)\geq2k-1$ almost every random lift of $G$ is $k$-linked.

scholarly journals The Chromatic Index of a Graph Whose Core has Maximum Degree $2$

10.37236/2101 ◽  
2012 ◽  
Vol 19 (1) ◽  
Author(s):  
Mikio Kano ◽  
Saieed Akbari ◽  
Maryam Ghanbari ◽  
Mohammad Javad Nikmehr
Keyword(s):  
Connected Graph ◽  
Maximum Degree ◽  
Edge Coloring ◽  
Chromatic Index ◽  
The Core ◽  
Class 1 ◽  
Minimum Number ◽  
Even Order ◽  
Delta G

Let $G$ be a graph. The core of $G$, denoted by $G_{\Delta}$, is the subgraph of $G$ induced by the vertices of degree $\Delta(G)$, where $\Delta(G)$ denotes the maximum degree of $G$. A $k$-edge coloring of $G$ is a function $f:E(G)\rightarrow L$ such that $|L| = k$ and $f(e_1)\neq f(e_2)$ for all two adjacent edges  $e_1$ and $e_2$ of $G$. The chromatic index of $G$, denoted by $\chi'(G)$, is the minimum number $k$ for which $G$ has a $k$-edge coloring.  A graph $G$ is said to be Class $1$ if $\chi'(G) = \Delta(G)$ and Class $2$ if $\chi'(G) = \Delta(G) + 1$. In this paper it is shown that every connected graph $G$ of even order and with $\Delta(G_{\Delta})\leq 2$ is Class $1$ if $|G_{\Delta}|\leq 9$ or $G_{\Delta}$ is a cycle of order $10$.

scholarly journals Containment Game Played on Random Graphs: Another Zig-Zag Theorem

10.37236/4777 ◽  
2015 ◽  
Vol 22 (2) ◽  
Author(s):  
Paweł Prałat
Keyword(s):  
Random Graphs ◽  
Maximum Degree ◽  
Domination Number ◽  
Main Question ◽  
Wide Range ◽  
Minimum Number ◽  
Natural Counterpart ◽  
Delta G ◽  
The Moment ◽  
Zigzag Shape

We consider a variant of the game of Cops and Robbers, called Containment, in which cops move from edge to adjacent edge, the robber moves from vertex to adjacent vertex (but cannot move along an edge occupied by a cop). The cops win by "containing'' the robber, that is, by occupying all edges incident with a vertex occupied by the robber. The minimum number of cops, $\xi(G)$, required to contain a robber played on a graph $G$ is called the containability number, a natural counterpart of the well-known cop number $c(G)$. This variant of the game was recently introduced by Komarov and Mackey, who proved that for every graph $G$, $c(G) \le \xi(G) \le \gamma(G) \Delta(G)$, where $\gamma(G)$ and $\Delta(G)$ are the domination number and the maximum degree of $G$, respectively. They conjecture that an upper bound can be improved and, in fact, $\xi(G) \le c(G) \Delta(G)$. (Observe that, trivially, $c(G) \le \gamma(G)$.) This seems to be the main question for this game at the moment. By investigating expansion properties, we provide asymptotically almost sure bounds on the containability number of binomial random graphs $\mathcal{G}(n,p)$ for a wide range of $p=p(n)$, showing that it forms an intriguing zigzag shape. This result also proves that the conjecture holds for some range of $p$ (or holds up to a constant or an $O(\log n)$ multiplicative factors for some other ranges).

scholarly journals Effects of Structural Parameters on Seismic Behaviour of Historical Masonry Minaret

2017 ◽  
Author(s):  
Barış Erdil ◽  
Mücip Tapan ◽  
İsmail Akkaya ◽  
Fuat Korkut
Keyword(s):  
Modal Analysis ◽  
Structural Parameters ◽  
Masonry Structure ◽  
Operational Modal Analysis ◽  
Shear Cracks ◽  
Seismic Behaviour ◽  
New Model ◽  
The Core ◽  
Historical Masonry ◽  
Stiffness Distribution

The October 23, 2011 (Mw = 7.2) and November 9, 2011 (Mw = 5.6) earthquakes increased the damage in the minaret of Van Ulu Mosque, an important historical masonry structure built with solid bricks in Eastern Turkey, resulting in significant shear cracks. It was found that since the door and window openings are not symmetrically placed, they result in unsymmetrical stiffness distribution. The contribution of staircase and the core on stiffness is ignorable but its effect on the mass is significant. The pulpit with chamfered corner results in unsymmetrical transverse displacements. Brace wall improves the stiffness however contributes to the unsymmetrical behaviour considerably. The reason for the diagonal cracks can be attributed to the unsymmetrical brace wall and the chamfered pulpit but the effect of brace wall is more pronounced. After introducing the cracks, a new model was created and calibrated according to the results of Operational Modal Analysis. Diagonal cracks were found to be likely to develop under earthquake loading. Drifts are observed to increase significantly upon the introduction of the cracks.

scholarly journals A Strengthening of Brooks' Theorem for Line Graphs

10.37236/632 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Landon Rabern
Keyword(s):  
Chromatic Number ◽  
Maximum Degree ◽  
Line Graph ◽  
Clique Number ◽  
Line Graphs ◽  
Delta G

We prove that if $G$ is the line graph of a multigraph, then the chromatic number $\chi(G)$ of $G$ is at most $\max\left\{\omega(G), \frac{7\Delta(G) + 10}{8}\right\}$ where $\omega(G)$ and $\Delta(G)$ are the clique number and the maximum degree of $G$, respectively. Thus Brooks' Theorem holds for line graphs of multigraphs in much stronger form. Using similar methods we then prove that if $G$ is the line graph of a multigraph with $\chi(G) \geq \Delta(G) \geq 9$, then $G$ contains a clique on $\Delta(G)$ vertices. Thus the Borodin-Kostochka Conjecture holds for line graphs of multigraphs.

scholarly journals Vizing's 2-Factor Conjecture Involving Toughness and Maximum Degree Conditions

10.37236/7353 ◽  
2019 ◽  
Vol 26 (2) ◽  
Author(s):  
Jinko Kanno ◽  
Songling Shan
Keyword(s):  
Maximum Degree ◽  
Hamiltonian Cycle ◽  
Simple Graph ◽  
Chromatic Index ◽  
New Techniques ◽  
Critical Graph ◽  
Degree Conditions ◽  
Delta G ◽  
Proper Subgraph

Let $G$ be a simple graph, and let $\Delta(G)$ and $\chi'(G)$ denote the maximum degree and chromatic index of $G$, respectively. Vizing proved that $\chi'(G)=\Delta(G)$ or $\chi'(G)=\Delta(G)+1$. We say $G$ is $\Delta$-critical if $\chi'(G)=\Delta(G)+1$ and $\chi'(H)<\chi'(G)$ for every proper subgraph $H$ of $G$. In 1968, Vizing conjectured that if $G$ is a $\Delta$-critical graph, then  $G$ has a 2-factor. Let $G$ be an $n$-vertex $\Delta$-critical graph. It was proved that if $\Delta(G)\ge n/2$, then $G$ has a 2-factor; and that if $\Delta(G)\ge 2n/3+13$, then $G$  has a hamiltonian cycle, and thus a 2-factor. It is well known that every 2-tough graph with at least three vertices has a 2-factor. We investigate the existence of a 2-factor in a $\Delta$-critical graph under "moderate" given toughness and  maximum degree conditions. In particular, we show that  if $G$ is an  $n$-vertex $\Delta$-critical graph with toughness at least 3/2 and with maximum degree at least $n/3$, then $G$ has a 2-factor. We also construct a family of graphs that have order $n$, maximum degree $n-1$, toughness at least $3/2$, but have no 2-factor. This implies that the $\Delta$-criticality in the result is needed. In addition, we develop new techniques in proving the existence of 2-factors in graphs.

Pitchfork domination in graphs

2020 ◽  
Vol 12 (02) ◽  
pp. 2050025
Author(s):  
Manal N. Al-Harere ◽  
Mohammed A. Abdlhusein
Keyword(s):  
Maximum Degree ◽  
Undirected Graph ◽  
Minimum Degree ◽  
Dominating Set ◽  
Domination Number ◽  
Dominating Sets ◽  
Minimum Cardinality ◽  
New Model ◽  
Domination In Graphs

In this paper, a new model of domination in graphs called the pitchfork domination is introduced. Let [Formula: see text] be a finite, simple and undirected graph without isolated vertices, a subset [Formula: see text] of [Formula: see text] is a pitchfork dominating set if every vertex [Formula: see text] dominates at least [Formula: see text] and at most [Formula: see text] vertices of [Formula: see text], where [Formula: see text] and [Formula: see text] are non-negative integers. The domination number of [Formula: see text], denotes [Formula: see text] is a minimum cardinality over all pitchfork dominating sets in [Formula: see text]. In this work, pitchfork domination when [Formula: see text] and [Formula: see text] is studied. Some bounds on [Formula: see text] related to the order, size, minimum degree, maximum degree of a graph and some properties are given. Pitchfork domination is determined for some known and new modified graphs. Finally, a question has been answered and discussed that; does every finite, simple and undirected graph [Formula: see text] without isolated vertices have a pitchfork domination or not?

scholarly journals On Komlós’ tiling theorem in random graphs

2019 ◽  
Vol 29 (1) ◽  
pp. 113-127
Author(s):  
Rajko Nenadov ◽  
Nemanja Škorić
Keyword(s):  
Chromatic Number ◽  
Random Graph ◽  
Random Graphs ◽  
Upper Bound ◽  
Minimum Degree ◽  
Arbitrary Graph ◽  
Spanning Subgraph ◽  
Delta G ◽  
Image Position ◽  
Vertex Disjoint

AbstractGiven graphs G and H, a family of vertex-disjoint copies of H in G is called an H-tiling. Conlon, Gowers, Samotij and Schacht showed that for a given graph H and a constant γ>0, there exists C>0 such that if $p \ge C{n^{ - 1/{m_2}(H)}}$ , then asymptotically almost surely every spanning subgraph G of the random graph 𝒢(n, p) with minimum degree at least $\delta (G) \ge (1 - \frac{1}{{{\chi _{{\rm{cr}}}}(H)}} + \gamma )np$ contains an H-tiling that covers all but at most γn vertices. Here, χcr(H) denotes the critical chromatic number, a parameter introduced by Komlós, and m2(H) is the 2-density of H. We show that this theorem can be bootstrapped to obtain an H-tiling covering all but at most $\gamma {(C/p)^{{m_2}(H)}}$ vertices, which is strictly smaller when $p \ge C{n^{ - 1/{m_2}(H)}}$ . In the case where H = K3, this answers the question of Balogh, Lee and Samotij. Furthermore, for an arbitrary graph H we give an upper bound on p for which some leftover is unavoidable and a bound on the size of a largest H -tiling for p below this value.

On Random Intersection Graphs: The Subgraph Problem

1999 ◽  
Vol 8 (1-2) ◽  
pp. 131-159 ◽  
Author(s):  
MICHAŁ KAROŃSKI ◽  
EDWARD R. SCHEINERMAN ◽  
KAREN B. SINGER-COHEN
Keyword(s):  
Circuit Design ◽  
Random Graphs ◽  
Induced Subgraphs ◽  
Intersection Graphs ◽  
New Model ◽  
Gate Matrix

A new model of random graphs – random intersection graphs – is introduced. In this model, vertices are assigned random subsets of a given set. Two vertices are adjacent provided their assigned sets intersect. We explore the evolution of random intersection graphs by studying thresholds for the appearance and disappearance of small induced subgraphs. An application to gate matrix circuit design is presented.

scholarly journals A spanning bandwidth theorem in random graphs

2021 ◽  
pp. 1-31
Author(s):  
Peter Allen ◽  
Julia Böttcher ◽  
Julia Ehrenmüller ◽  
Jakob Schnitzer ◽  
Anusch Taraz
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Maximum Degree ◽  
Minimum Degree ◽  
Spanning Subgraph ◽  
Vertex Graph ◽  
Special Case

Abstract The bandwidth theorem of Böttcher, Schacht and Taraz states that any n-vertex graph G with minimum degree $\big(\tfrac{k-1}{k}+o(1)\big)n$ contains all n-vertex k-colourable graphs H with bounded maximum degree and bandwidth o(n). Recently, a subset of the authors proved a random graph analogue of this statement: for $p\gg \big(\tfrac{\log n}{n}\big)^{1/\Delta}$ a.a.s. each spanning subgraph G of G(n,p) with minimum degree $\big(\tfrac{k-1}{k}+o(1)\big)pn$ contains all n-vertex k-colourable graphs H with maximum degree $\Delta$ , bandwidth o(n), and at least $C p^{-2}$ vertices not contained in any triangle. This restriction on vertices in triangles is necessary, but limiting. In this paper, we consider how it can be avoided. A special case of our main result is that, under the same conditions, if additionally all vertex neighbourhoods in G contain many copies of $K_\Delta$ then we can drop the restriction on H that $Cp^{-2}$ vertices should not be in triangles.

scholarly journals Differentials in certain classes of graphs

2010 ◽  
Vol 41 (2) ◽  
pp. 129-138 ◽  
Author(s):  
P. Roushini Leely Pushpam ◽  
D. Yokesh
Keyword(s):  
Binary Tree ◽  
Maximum Degree ◽  
Bipartite Graphs ◽  
Regular Graphs ◽  
Complete Binary Tree ◽  
Unicyclic Graphs ◽  
Grid Graphs ◽  
Split Graphs ◽  
Delta G

Let $X subset V$ be a set of vertices in a graph $G = (V, E)$. The boundary $B(X)$ of $X$ is defined to be the set of vertices in $V-X$ dominated by vertices in $X$, that is, $B(X) = (V-X) cap N(X)$. The differential $ partial(X)$ of $X$ equals the value $ partial(X) = |B(X)| - |X|$. The differential of a graph $G$ is defined as $ partial(G) = max { partial(X) | X subset V }$. It is easy to see that for any graph $G$ having vertices of maximum degree $ Delta(G)$, $ partial(G) geq Delta (G) -1$. In this paper we characterize the classes of unicyclic graphs, split graphs, grid graphs, $k$-regular graphs, for $k leq 4$, and bipartite graphs for which $ partial(G) = Delta(G)-1$. We also determine the value of $ partial(T)$ for any complete binary tree $T$.

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