scholarly journals Complete graph immersions and minimum degree

2017 ◽  
Vol 88 (1) ◽  
pp. 211-221 ◽  
Author(s):  
Zdeněk Dvořák ◽  
Liana Yepremyan
Keyword(s):  
Complete Graph ◽  
Minimum Degree

scholarly journals A minimum degree condition forcing complete graph immersion

COMBINATORICA ◽  
2014 ◽  
Vol 34 (3) ◽  
pp. 279-298 ◽  
Author(s):  
Matt Devos ◽  
Zdeněk Dvořák ◽  
Jacob Fox ◽  
Jessica McDonald ◽  
Bojan Mohar ◽  
...  
Keyword(s):  
Complete Graph ◽  
Minimum Degree ◽  
Degree Condition

scholarly journals Monochromatic Components in Edge-Coloured Graphs with Large Minimum Degree

10.37236/9039 ◽  
2021 ◽  
Vol 28 (1) ◽  
Author(s):  
Hannah Guggiari ◽  
Alex Scott
Keyword(s):  
Complete Graph ◽  
Minimum Degree ◽  
Connected Component ◽  
Delta G ◽  
Edge Colouring

For every $n\in\mathbb{N}$ and $k\geqslant2$, it is known that every $k$-edge-colouring of the complete graph on $n$ vertices contains a monochromatic connected component of order at least $\frac{n}{k-1}$. For $k\geqslant3$, it is known that the complete graph can be replaced by a graph $G$ with $\delta(G)\geqslant(1-\varepsilon_k)n$ for some constant $\varepsilon_k$. In this paper, we show that the maximum possible value of $\varepsilon_3$ is $\frac16$. This disproves a conjecture of Gyárfas and Sárközy.

scholarly journals The Positive Minimum Degree Game on Sparse Graphs

10.37236/1174 ◽  
2012 ◽  
Vol 19 (1) ◽  
Author(s):  
József Balogh ◽  
András Pluhár
Keyword(s):  
Complete Graph ◽  
Special Form ◽  
Minimum Degree ◽  
Sharp Bound ◽  
Sparse Graphs ◽  
Discharging Method ◽  
Graph Game

In this note we investigate a special form of degree games defined by D. Hefetz, M. Krivelevich, M. Stojaković and T. Szabó. Usually the board of a graph game is the edge set of $K_n$, the complete graph on $n$ vertices. Maker and Breaker alternately claim an edge, and Maker wins if his edges form a subgraph with prescribed properties; here a certain minimum degree. In the special form the board is no longer the whole edge set of $K_n$, Maker first selects as few edges of $K_n$ as possible in order to win, and our goal is to compute the necessary size of that board. Solving a question of Hefetz et al., we show, using the discharging method, that the sharp bound is around $10n/7$ for the positive minimum degree game.

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 $2t+r$ .

Ramsey Problems with Bounded Degree Spread

1993 ◽  
Vol 2 (3) ◽  
pp. 263-269 ◽  
Author(s):  
G. Chen ◽  
R. H. Schelp
Keyword(s):  
Positive Integer ◽  
Complete Graph ◽  
Maximum Degree ◽  
Minimum Degree ◽  
Bipartite Graphs ◽  
Bounded Degree ◽  
Monochromatic Copy

Let k be a positive integer, k ≥ 2. In this paper we study bipartite graphs G such that, for n sufficiently large, each two-coloring of the edges of the complete graph Kn gives a monochromatic copy of G, with some k of its vertices having the maximum degree of these k vertices minus the minimum degree of these k vertices (in the colored Kn) at most k − 2.

scholarly journals Induced Complete $h$-partite Graphs in Dense Clique-less Graphs

10.37236/1475 ◽  
1999 ◽  
Vol 6 (1) ◽  
Author(s):  
Eldar Fischer
Keyword(s):  
Complete Graph ◽  
Minimum Degree ◽  
Color Class ◽  
Vertex Disjoint

It is proven that for every fixed $h$, $a$ and $b$, a graph with $n$ vertices and minimum degree at least ${h-1 \over h}n$, which contains no copy of $K_b$ (the complete graph with $b$ vertices), contains at least $(1-o(1)){n \over ha}$ vertex disjoint induced copies of the complete $h$-partite graph with $a$ vertices in each color class.

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.

On Dirac's Generalization of Brooks' Theorem

1972 ◽  
Vol 24 (5) ◽  
pp. 805-807 ◽  
Author(s):  
Hudson V. Kronk ◽  
John Mitchem
Keyword(s):  
Chromatic Number ◽  
Complete Graph ◽  
Connected Graph ◽  
Maximum Degree ◽  
Minimum Degree ◽  
Equivalent Formulation ◽  
Critical Graphs ◽  
Critical Graph ◽  
Odd Cycle ◽  
Proper Subgraph

It is easy to verify that any connected graph G with maximum degree s has chromatic number χ(G) ≦ 1 + s. In [1], R. L. Brooks proved that χ(G) ≦ s, unless s = 2 and G is an odd cycle or s > 2 and G is the complete graph Ks+1. This was the first significant theorem connecting the structure of a graph with its chromatic number. For s ≦ 4, Brooks' theorem says that every connected s-chromatic graph other than Ks contains a vertex of degree > s — 1. An equivalent formulation can be given in terms of s-critical graphs. A graph G is said to be s-critical if χ(G) = s, but every proper subgraph has chromatic number less than s. Each scritical graph has minimum degree ≦ s — 1. We can now restate Brooks' theorem: if an s-critical graph, s ≦ 4, is not Ks and has p vertices and q edges, then 2q ≦ (s — l)p + 1. Dirac [2] significantly generalized the theorem of Brooks by showing that 2q ≦ (s — 1)£ + s — 3 and that this result is best possible.

scholarly journals Large Monochromatic Components in Edge Colored Graphs with a Minimum Degree Condition

10.37236/7049 ◽  
2017 ◽  
Vol 24 (3) ◽  
Author(s):  
András Gyárfás ◽  
Gábor Sárközy
Keyword(s):  
Complete Graph ◽  
Minimum Degree ◽  
Complete Graphs ◽  
Connected Component ◽  
Degree Condition ◽  
Colored Graphs ◽  
Spanning Subgraph ◽  
Delta G

It is well-known that in every $k$-coloring of the edges of the complete graph $K_n$ there is a monochromatic connected component of order at least ${n\over k-1}$. In this paper we study an extension of this problem by replacing complete graphs by graphs of large minimum degree. For $k=2$ the authors proved that $\delta(G)\ge{3n\over 4}$ ensures a monochromatic connected component with at least $\delta(G)+1$ vertices in every $2$-coloring of the edges of a graph $G$ with $n$ vertices. This result is sharp, thus for $k=2$ we really need a complete graph to guarantee that one of the colors has a monochromatic connected spanning subgraph. Our main result here is  that for larger values of $k$ the situation is different, graphs of minimum degree $(1-\epsilon_k)n$ can replace complete graphs and still there is a monochromatic connected component of order at least ${n\over k-1}$, in fact $$\delta(G)\ge \left(1 - \frac{1}{1000(k-1)^9}\right)n$$ suffices.Our second result is an improvement of this bound for $k=3$. If the edges of $G$ with  $\delta(G)\geq {9n\over 10}$ are $3$-colored, then there is a monochromatic component of order at least ${n\over 2}$. We conjecture that this can be improved to ${7n\over 9}$ and for general $k$ we conjecture the following: if $k\geq 3$ and  $G$ is a graph of order $n$ such that $\delta(G)\geq \left( 1 - \frac{k-1}{k^2}\right)n$, then in any $k$-coloring of the edges of $G$ there is a monochromatic connected component of order at least ${n\over k-1}$.

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