scholarly journals Random Subgraphs in Cartesian Powers of Regular Graphs

10.37236/2058 ◽  
2012 ◽  
Vol 19 (1) ◽  
Author(s):  
Felix Joos
Keyword(s):  
Regular Graph ◽  
Regular Graphs ◽  
Cartesian Power ◽  
Random Subgraph ◽  
Random Subgraphs

Let $G$ be a connected $d$-regular graph with $k$ vertices. We investigate the behaviour of a spanning random subgraph $G^n_p$ of $G^n$, the $n$-th Cartesian power of $G$, which is constructed by deleting each edge independently with probability $1-p$. We prove that $\lim\limits_{n \rightarrow \infty} \mathbb{P}[G^n_p {\rm \ is \ connected}]=e^{-\lambda}$, if $p=p(n)=1-\left(\frac{\lambda_n^{1/n}}{k}\right)^{1/d}$ and $\lambda_n \rightarrow \lambda>0$ as $n \rightarrow \infty$. This extends a result of L. Clark, Random subgraphs of certain graph powers, Int. J. Math. Math. Sci., 32(5):285-292, 2002.

scholarly journals Ramsey Properties of Random Subgraphs of Pseudo-Random Graphs

10.37236/2468 ◽  
2012 ◽  
Vol 19 (3) ◽  
Author(s):  
Jia Shen
Keyword(s):  
Random Graphs ◽  
Regular Graph ◽  
Threshold Function ◽  
Vertex Coloring ◽  
Similar Argument ◽  
Sharp Threshold ◽  
Absolute Value ◽  
Random Subgraph ◽  
Vertex Disjoint ◽  
Random Subgraphs

Let $G=(V,E)$ be a $d$-regular graph of order $n$. Let $G_p$ be the random subgraph of $G$ for which each edge is selected from $E(G)$ independently at random with probability $p$. For a fixed graph $H$, define $m(H):=$max$\{e(H')/(v(H')-1):H' \subseteq H\}$. We prove that $n^{(m(H)-1)/m(H)}/d$ is a threshold function for $G_p$ to satisfy Ramsey, induced Ramsey, and canonical Ramsey properties with respect to vertex coloring, respectively, provided the eigenvalue $\lambda$ of $G$ that is second largest in absolute value is significantly smaller than $d$.As a consequence, it is also shown that $\displaystyle n^{(m(H)-1)/m(H)}/d$ is a threshold function for $G_p$ to contain a family of vertex disjoint copies of $H$ (an $H$ packing) that covers $(1-o(1))n$ vertices of $G$. Using a similar argument, the sharp threshold function for $G_p$ to contain $H$ as a subgraph is obtained as well.

scholarly journals Random subgraphs of certain graph powers

2002 ◽  
Vol 32 (5) ◽  
pp. 285-292 ◽  
Author(s):  
Lane Clark
Keyword(s):  
Cartesian Power ◽  
Random Subgraph ◽  
Random Subgraphs

We determine the limiting probability that a random subgraph of the Cartesian powerKanor ofKa,anis connected.

Extreme degrees in random subgraphs of regular graphs

1985 ◽  
Vol 97 (1) ◽  
pp. 69-78 ◽  
Author(s):  
Zbigniew Palka
Keyword(s):  
Minimum Degree ◽  
Probability Distributions ◽  
Asymptotic Distributions ◽  
Regular Graphs ◽  
Additional Information ◽  
Initial Graph ◽  
Random Subgraph ◽  
Edge Probability ◽  
Maximum And Minimum Degree ◽  
Random Subgraphs

Let G(d) be a given simple d-regular graph on n labelled vertices, where dn is even. Such a graph will be called an initial graph. Denote by Gp(d) a random subgraph of G(d) obtained by removing edges, each with the same probability q — 1 —p, independently of all other edges (i.e. each edge remains in Gp(d) with probability p). In a recent paper [10] the asymptotic distributions of the number of vertices of a given degree in a random graph Gp(d) were given. The aim of this sequel is to present a wide variety of results devoted to probability distributions of the maximum and minimum degree of Gp(d) with respect to different values of the edge probability p and degree of regularity d. It should be noted here that very detailed results on a similar subject in the case when the initial graph is a complete graph (i.e. when d = n – 1) have already been obtained by Bollobás in the series of papers [2]–[4] (some additional information to the paper [4] was given in [9]). Also, in proving our results we will make use of some ideas given by Bollobás in these papers.

scholarly journals Cycle partitions of regular graphs

2020 ◽  
pp. 1-24
Author(s):  
Vytautas Gruslys ◽  
Shoham Letzter
Keyword(s):  
Regular Graph ◽  
Bipartite Graphs ◽  
Disjoint Paths ◽  
Regular Graphs ◽  
Simple Construction ◽  
Vertex Set ◽  
Almost All ◽  
Vertex Disjoint

Abstract Magnant and Martin conjectured that the vertex set of any d-regular graph G on n vertices can be partitioned into $n / (d+1)$ paths (there exists a simple construction showing that this bound would be best possible). We prove this conjecture when $d = \Omega(n)$ , improving a result of Han, who showed that in this range almost all vertices of G can be covered by $n / (d+1) + 1$ vertex-disjoint paths. In fact our proof gives a partition of V(G) into cycles. We also show that, if $d = \Omega(n)$ and G is bipartite, then V(G) can be partitioned into n/(2d) paths (this bound is tight for bipartite graphs).

The maximum number of vertices of primitive regular graphs of orders 2, 3, 4 with exponent 2

2021 ◽  
pp. 97-104
Author(s):  
M. B. Abrosimov ◽  
◽  
S. V. Kostin ◽  
I. V. Los ◽  
◽  
...  
Keyword(s):  
Regular Graph ◽  
Regular Graphs ◽  
Similar Question

In 2015, the results were obtained for the maximum number of vertices nk in regular graphs of a given order k with a diameter 2: n2 = 5, n3 = 10, n4 = 15. In this paper, we investigate a similar question about the largest number of vertices npk in a primitive regular graph of order k with exponent 2. All primitive regular graphs with exponent 2, except for the complete one, also have diameter d = 2. The following values were obtained for primitive regular graphs with exponent 2: np2 = 3, np3 = 4, np4 = 11.

Minimal Regular Graphs of Girths Eight and Twelve

1966 ◽  
Vol 18 ◽  
pp. 1091-1094 ◽  
Author(s):  
Clark T. Benson
Keyword(s):  
Group Theory ◽  
Lower Bound ◽  
Regular Graph ◽  
Regular Graphs ◽  
Image Position

In (3) Tutte showed that the order of a regular graph of degree d and even girth g > 4 is greater than or equal toHere the girth of a graph is the length of the shortest circuit. It was shown in (2) that this lower bound cannot be attained for regular graphs of degree > 2 for g ≠ 6, 8, or 12. When this lower bound is attained, the graph is called minimal. In a group-theoretic setting a similar situation arose and it was noticed by Gleason that minimal regular graphs of girth 12 could be constructed from certain groups. Here we construct these graphs making only incidental use of group theory. Also we give what is believed to be an easier construction of minimal regular graphs of girth 8 than is given in (2). These results are contained in the following two theorems.

scholarly journals On the Number of Spanning Trees in Random Regular Graphs

10.37236/3752 ◽  
2014 ◽  
Vol 21 (1) ◽  
Author(s):  
Catherine Greenhill ◽  
Matthew Kwan ◽  
David Wind
Keyword(s):  
Asymptotic Distribution ◽  
Regular Graph ◽  
Spanning Trees ◽  
Regular Graphs ◽  
Cubic Graph ◽  
Expected Number ◽  
Fixed Integer ◽  
Numerical Evidence ◽  
Number Of Spanning Trees

Let $d\geq 3$ be a fixed integer.   We give an asympotic formula for the expected number of spanning trees in a uniformly random $d$-regular graph with $n$ vertices. (The asymptotics are as $n\to\infty$, restricted to even $n$ if $d$ is odd.) We also obtain the asymptotic distribution of the number of spanning trees in a uniformly random cubic graph, and conjecture that the corresponding result holds for arbitrary (fixed) $d$. Numerical evidence is presented which supports our conjecture.

scholarly journals On Regular Factors in Regular Graphs with Small Radius

10.37236/1760 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
Arne Hoffmann ◽  
Lutz Volkmann
Keyword(s):  
Regular Graph ◽  
Small Radius ◽  
Regular Graphs ◽  
High Eccentricity ◽  
K Factor

In this note we examine the connection between vertices of high eccentricity and the existence of $k$-factors in regular graphs. This leads to new results in the case that the radius of the graph is small ($\leq 3$), namely that a $d$-regular graph $G$ has all $k$-factors, for $k|V(G)|$ even and $k\le d$, if it has at most $2d+2$ vertices of eccentricity $>3$. In particular, each regular graph $G$ of diameter $\leq3$ has every $k$-factor, for $k|V(G)|$ even and $k\le d$.

scholarly journals Smallest regular graphs without near 1-factors

1986 ◽  
Vol 41 (2) ◽  
pp. 193-210 ◽  
Author(s):  
Gary Chartrand ◽  
Sergio Ruiz ◽  
Curtiss E. Wall
Keyword(s):  
Regular Graph ◽  
Connected Graph ◽  
Regular Graphs ◽  
Cubic Graph ◽  
Subgraph Isomorphic ◽  
Even Order

AbstractA near 1-factor of a graph of order 2n ≧ 4 is a subgraph isomorphic to (n − 2) K2 ∪ P3 ∪ K1. Wallis determined, for each r ≥ 3, the order of a smallest r-regular graph of even order without a 1-factor; while for each r ≧ 3, Chartrand, Goldsmith and Schuster determined the order of a smallest r-regular, (r − 2)-edge-connected graph of even order without a 1-factor. These results are extended to graphs without near 1-factors. It is known that every connected, cubic graph with less than six bridges has a near 1-factor. The order of a smallest connected, cubic graph with exactly six bridges and no near 1-factor is determined.

On the Non-Existence of a Type of Regular Graphs of Girth 5

1967 ◽  
Vol 19 ◽  
pp. 644-648 ◽  
Author(s):  
William G. Brown
Keyword(s):  
Lower Bound ◽  
Regular Graph ◽  
Regular Graphs ◽  
Minimal Graph ◽  
Image Position

ƒ(k, 5) is defined to be the smallest integer n for which there exists a regular graph of valency k and girth 5, having n vertices. In (3) it was shown that1.1Hoffman and Singleton proved in (4) that equality holds in the lower bound of (1.1) only for k = 2, 3, 7, and possibly 57. Robertson showed in (6) that ƒ(4, 5) = 19 and constructed the unique minimal graph.

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