scholarly journals Partitioning Random Graphs into Monochromatic Components

10.37236/6089 ◽  
2017 ◽  
Vol 24 (1) ◽  
Author(s):  
Deepak Bal ◽  
Louis DeBiasio
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Complete Graph ◽  
Minimum Degree ◽  
Partial Result ◽  
Complete Graphs ◽  
Number Of Components ◽  
Classic Result ◽  
Bounded Number

Erdős, Gyárfás, and Pyber (1991) conjectured that every $r$-colored complete graph can be partitioned into at most $r-1$ monochromatic components; this is a strengthening of a conjecture of Lovász (1975) and Ryser (1970) in which the components are only required to form a cover. An important partial result of Haxell and Kohayakawa (1995) shows that a partition into $r$ monochromatic components is possible for sufficiently large $r$-colored complete graphs.We start by extending Haxell and Kohayakawa's result to graphs with large minimum degree, then we provide some partial analogs of their result for random graphs. In particular, we show that if $p\ge \left(\frac{27\log n}{n}\right)^{1/3}$, then a.a.s. in every $2$-coloring of $G(n,p)$ there exists a partition into two monochromatic components, and for $r\geq 2$ if $p\ll \left(\frac{r\log n}{n}\right)^{1/r}$, then a.a.s. there exists an $r$-coloring of $G(n,p)$ such that there does not exist a cover with a bounded number of components. Finally, we consider a random graph version of a classic result of Gyárfás (1977) about large monochromatic components in $r$-colored complete graphs. We show that if $p=\frac{\omega(1)}{n}$, then a.a.s. in every $r$-coloring of $G(n,p)$ there exists a monochromatic component of order at least $(1-o(1))\frac{n}{r-1}$.

scholarly journals The Threshold Probability for Long Cycles

2016 ◽  
Vol 26 (2) ◽  
pp. 208-247 ◽  
Author(s):  
ROMAN GLEBOV ◽  
HUMBERTO NAVES ◽  
BENNY SUDAKOV
Keyword(s):  
Random Graph ◽  
Complete Graph ◽  
Minimum Degree ◽  
Hamilton Cycle ◽  
Threshold Probability ◽  
Spanning Subgraph ◽  
Classic Result ◽  
Long Cycles

For a given graph G of minimum degree at least k, let Gp denote the random spanning subgraph of G obtained by retaining each edge independently with probability p = p(k). We prove that if p ⩾ (logk + loglogk + ωk(1))/k, where ωk(1) is any function tending to infinity with k, then Gp asymptotically almost surely contains a cycle of length at least k + 1. When we take G to be the complete graph on k + 1 vertices, our theorem coincides with the classic result on the threshold probability for the existence of a Hamilton cycle in the binomial random graph.

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.

Some remarks about extreme degrees in a random graph

1986 ◽  
Vol 100 (1) ◽  
pp. 167-174 ◽  
Author(s):  
Zbigniew Palka
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Complete Graph ◽  
Probability Distributions ◽  
Supplementary Information ◽  
Random Subgraph

Let Kn, p be a random subgraph of a complete graph Kn obtained by removing edges, each with the same probability q = 1 – p, independently of all other edges (i.e. each edge remains in Kn, p with probability p). Very detailed results devoted to probability distributions of the number of vertices of a given degree, as well as of the extreme degrees of Kn, p, have already been obtained by many authors (see e.g. [l]–[5], [7]–[9]). A similar subject for other models of random graphs has been investigated in [10]–[13], The aim of this note is to give some supplementary information about the distribution of the ith smallest (i ≥ 1 is fixed) and the ith largest degree in a sparse random graph Kn, p, i.e. when p = p(n) = o(1).

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 Triangle-Free Subgraphs of Random Graphs

2017 ◽  
Vol 27 (2) ◽  
pp. 141-161
Author(s):  
PETER ALLEN ◽  
JULIA BÖTTCHER ◽  
YOSHIHARU KOHAYAKAWA ◽  
BARNABY ROBERTS
Keyword(s):  
Graph Theory ◽  
Random Graph ◽  
Random Graphs ◽  
Minimum Degree ◽  
Discrete Math ◽  
Constant Factor ◽  
Extremal Graph ◽  
Extremal Graph Theory ◽  
Spanning Subgraph

Recently there has been much interest in studying random graph analogues of well-known classical results in extremal graph theory. Here we follow this trend and investigate the structure of triangle-free subgraphs of G(n, p) with high minimum degree. We prove that asymptotically almost surely each triangle-free spanning subgraph of G(n, p) with minimum degree at least (2/5 + o(1))pn is (p−1n)-close to bipartite, and each spanning triangle-free subgraph of G(n, p) with minimum degree at least (1/3 + ϵ)pn is O(p−1n)-close to r-partite for some r = r(ϵ). These are random graph analogues of a result by Andrásfai, Erdős and Sós (Discrete Math.8 (1974), 205–218), and a result by Thomassen (Combinatorica22 (2002), 591–596). We also show that our results are best possible up to a constant factor.

Triangle Factors in Random Graphs

1997 ◽  
Vol 6 (3) ◽  
pp. 337-347 ◽  
Author(s):  
MICHAEL KRIVELEVICH
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
General Setting ◽  
Complete Graphs ◽  
Multiple Use ◽  
Vertex Disjoint

For a graph G=(V, E) on n vertices, where 3 divides n, a triangle factor is a subgraph of G, consisting of n/3 vertex disjoint triangles (complete graphs on three vertices). We discuss the problem of determining the minimal probability p=p(n), for which a random graph G∈[Gscr ](n, p) contains almost surely a triangle factor. This problem (in a more general setting) has been studied by Alon and Yuster and by Ruciński, their approach implies p=O((log n/n)1/2). Our main result is that p=O(n)−3/5) already suffices. The proof is based on a multiple use of the Janson inequality. Our approach can be extended to improve known results about the threshold for the existence of an H-factor in [Gscr ](n, p) for various graphs H.

scholarly journals Toppling numbers of complete and random graphs

2014 ◽  
Vol Vol. 16 no. 3 (Graph Theory) ◽  
Author(s):  
Anthony Bonato ◽  
William B. Kinnersley ◽  
Pawel Pralat
Keyword(s):  
Graph Theory ◽  
Differential Equations ◽  
Random Graphs ◽  
Complete Graph ◽  
Complete Graphs ◽  
Asymptotic Bounds ◽  
Large N ◽  
Chip Firing ◽  
International Audience ◽  
Person Game

Graph Theory International audience We study a two-person game played on graphs based on the widely studied chip-firing game. Players Max and Min alternately place chips on the vertices of a graph. When a vertex accumulates as many chips as its degree, it fires, sending one chip to each neighbour; this may in turn cause other vertices to fire. The game ends when vertices continue firing forever. Min seeks to minimize the number of chips played during the game, while Max seeks to maximize it. When both players play optimally, the length of the game is the toppling number of a graph G, and is denoted by t(G). By considering strategies for both players and investigating the evolution of the game with differential equations, we provide asymptotic bounds on the toppling number of the complete graph. In particular, we prove that for sufficiently large n 0.596400 n2 < t(Kn) < 0.637152 n2. Using a fractional version of the game, we couple the toppling numbers of complete graphs and the binomial random graph G(n,p). It is shown that for pn ≥n² / √ log(n) asymptotically almost surely t(G(n,p))=(1+o(1)) p t(Kn).

Cliques in random graphs

1976 ◽  
Vol 80 (3) ◽  
pp. 419-427 ◽  
Author(s):  
B. Bollobas ◽  
P. Erdös
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Complete Graph ◽  
Random Variables ◽  
Natural Numbers ◽  
Complete Subgraph ◽  
Maximal Size ◽  
Point Set ◽  
Image Position

Let 0 < p < 1 be fixed and denote by G a random graph with point set , the set of natural numbers, such that each edge occurs with probability p, independently of all other edges. In other words the random variables eij, 1 ≤ i < j, defined byare independent r.v.'s with P(eij = 1) = p, P(eij = 0) = 1 − p. Denote by Gn the subgraph of G spanned by the points 1, 2, …, n. These random graphs G, Gn will be investigated throughout the note. As in (1), denote by Kr a complete graph with r points and denote by kr(H) the number of Kr's in a graph H. A maximal complete subgraph is called a clique. In (1) one of us estimated the minimum of kr(H) provided H has n points and m edges. In this note we shall look at the random variablesthe number of Kr's in Gn, andthe maximal size of a clique in Gn.

scholarly journals Client-Waiter Games on Complete and Random Graphs

10.37236/6039 ◽  
2016 ◽  
Vol 23 (4) ◽  
Author(s):  
Oren Dean ◽  
Michael Krivelevich
Keyword(s):  
Positive Integer ◽  
Random Graph ◽  
Random Graphs ◽  
Complete Graph ◽  
Upper Bounds ◽  
Connected Component ◽  
Lower And Upper Bounds ◽  
Large Connected Component ◽  
Graph Property ◽  
Player Game

For a graph $ G $, a monotone increasing graph property $ \mathcal{P} $ and positive integer $ q $, we define the Client-Waiter game to be a two-player game which runs as follows. In each turn Waiter is offering Client a subset of at least one and at most $ q+1 $ unclaimed edges of $ G $ from which Client claims one, and the rest are claimed by Waiter. The game ends when all the edges have been claimed. If Client's graph has property $ \mathcal{P} $ by the end of the game, then he wins the game, otherwise Waiter is the winner. In this paper we study several Client-Waiter games on the edge set of the complete graph, and the $ H $-game on the edge set of the random graph. For the complete graph we consider games where Client tries to build a large star, a long path and a large connected component. We obtain lower and upper bounds on the critical bias for these games and compare them with the corresponding Waiter-Client games and with the probabilistic intuition. For the $ H $-game on the random graph we show that the known results for the corresponding Maker-Breaker game are essentially the same for the Client-Waiter game, and we extend those results for the biased games and for trees.

scholarly journals Local Resilience for Squares of Almost Spanning Cycles in Sparse Random Graphs

10.37236/6281 ◽  
2017 ◽  
Vol 24 (4) ◽  
Author(s):  
Andreas Noever ◽  
Angelika Steger
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Hamiltonian Cycle ◽  
Blow Up ◽  
Minimum Degree ◽  
Single Vertex ◽  
Long Cycle ◽  
Local Resilience ◽  
Delta G ◽  
Spanning Cycles

In 1962, Pósa conjectured that a graph $G=(V, E)$ contains a square of a Hamiltonian cycle if $\delta(G)\ge 2n/3$. Only more than thirty years later Komlós, Sárkőzy, and Szemerédi proved this conjecture using the so-called Blow-Up Lemma. Here we extend their result to a random graph setting. We show that for every $\epsilon > 0$ and $p=n^{-1/2+\epsilon}$ a.a.s. every subgraph of $G_{n,p}$ with minimum degree at least $(2/3+\epsilon)np$ contains the square of a cycle on $(1-o(1))n$ vertices. This is almost best possible in three ways: (1) for $p\ll n^{-1/2}$ the random graph will not contain any square of a long cycle (2) one cannot hope for a resilience version for the square of a spanning cycle (as deleting all edges in the neighborhood of single vertex destroys this property) and (3) for $c<2/3$ a.a.s. $G_{n,p}$ contains a subgraph with minimum degree at least $cnp$ which does not contain the square of a path on $(1/3+c)n$ vertices.

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