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 ${n\over k-1}$.

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.

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 A New Bound on the Domination Number of Graphs with Minimum Degree Two

10.37236/499 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Michael A. Henning ◽  
Ingo Schiermeyer ◽  
Anders Yeo
Keyword(s):  
Connected Graph ◽  
Minimum Degree ◽  
Domination Number ◽  
Degree Condition ◽  
Cut Vertex ◽  
Delta G ◽  
Vertex Disjoint ◽  
Special Cycle

For a graph $G$, let $\gamma(G)$ denote the domination number of $G$ and let $\delta(G)$ denote the minimum degree among the vertices of $G$. A vertex $x$ is called a bad-cut-vertex of $G$ if $G-x$ contains a component, $C_x$, which is an induced $4$-cycle and $x$ is adjacent to at least one but at most three vertices on $C_x$. A cycle $C$ is called a special-cycle if $C$ is a $5$-cycle in $G$ such that if $u$ and $v$ are consecutive vertices on $C$, then at least one of $u$ and $v$ has degree $2$ in $G$. We let ${\rm bc}(G)$ denote the number of bad-cut-vertices in $G$, and ${\rm sc}(G)$ the maximum number of vertex disjoint special-cycles in $G$ that contain no bad-cut-vertices. We say that a graph is $(C_4,C_5)$-free if it has no induced $4$-cycle or $5$-cycle. Bruce Reed [Paths, stars and the number three. Combin. Probab. Comput. 5 (1996), 277–295] showed that if $G$ is a graph of order $n$ with $\delta(G) \ge 3$, then $\gamma(G) \le 3n/8$. In this paper, we relax the minimum degree condition from three to two. Let $G$ be a connected graph of order $n \ge 14$ with $\delta(G) \ge 2$. As an application of Reed's result, we show that $\gamma(G) \le \frac{1}{8} ( 3n + {\rm sc}(G) + {\rm bc}(G))$. As a consequence of this result, we have that (i) $\gamma(G) \le 2n/5$; (ii) if $G$ contains no special-cycle and no bad-cut-vertex, then $\gamma(G) \le 3n/8$; (iii) if $G$ is $(C_4,C_5)$-free, then $\gamma(G) \le 3n/8$; (iv) if $G$ is $2$-connected and $d_G(u) + d_G(v) \ge 5$ for every two adjacent vertices $u$ and $v$, then $\gamma(G) \le 3n/8$. All bounds are sharp.

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.

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 Large Monochromatic Components in Almost Complete Graphs and Bipartite Graphs

10.37236/9824 ◽  
2021 ◽  
Vol 28 (2) ◽  
Author(s):  
Zoltán Füredi ◽  
Ruth Luo
Keyword(s):  
Minimum Degree ◽  
Edge Coloring ◽  
Bipartite Graphs ◽  
Complete Graphs ◽  
Connected Component ◽  
Vertex Graph

Gyárfas proved that every coloring of the edges of $K_n$ with $t+1$ colors contains a monochromatic connected component of size at least $n/t$. Later, Gyárfás and Sárközy asked for which values of $\gamma=\gamma(t)$ does the following strengthening for almost complete graphs hold: if $G$ is an $n$-vertex graph with minimum degree at least $(1-\gamma)n$, then every $(t+1)$-edge coloring of $G$ contains a monochromatic component of size at least $n/t$. We show $\gamma= 1/(6t^3)$ suffices, improving a result of DeBiasio, Krueger, and Sárközy.

scholarly journals Rainbow Matchings of Size $\delta(G)$ in Properly Edge-Colored Graphs

10.37236/2443 ◽  
2012 ◽  
Vol 19 (2) ◽  
Author(s):  
Jennifer Diemunsch ◽  
Michael Ferrara ◽  
Allan Lo ◽  
Casey Moffatt ◽  
Florian Pfender ◽  
...  
Keyword(s):  
Minimum Degree ◽  
Time Algorithm ◽  
Colored Graph ◽  
Colored Graphs ◽  
Rainbow Matching ◽  
Delta G

A rainbow matching in an edge-colored graph is a matching in which all the edges have distinct colors. Wang asked if there is a function $f(\delta)$ such that a properly edge-colored graph $G$ with minimum degree $\delta$ and order at least $f(\delta)$ must have a rainbow matching of size $\delta$. We answer this question in the affirmative; an extremal approach yields that $f(\delta) = 98\delta/23< 4.27\delta$ suffices. Furthermore, we give an $O(\delta(G)|V(G)|^2)$-time algorithm that generates such a matching in a properly edge-colored graph of order at least $6.5\delta$. 

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 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 Generalized Hadamard Matrices and 2-Factorization of Complete Graphs

2020 ◽  
pp. 144-151
Author(s):  
W. V. Nishadi ◽  
A. A. I. Perera
Keyword(s):  
Cyclic Group ◽  
Complete Graph ◽  
Hadamard Matrix ◽  
Edge Coloring ◽  
Hadamard Matrices ◽  
Hamiltonian Cycles ◽  
Complete Graphs ◽  
Spanning Subgraph ◽  
Generalized Hadamard Matrix ◽  
Conjugate Transpose

Graph factorization plays a major role in graph theory and it shares common ideas in important problems such as edge coloring and Hamiltonian cycles. A factor  of a graph  is a spanning subgraph of  which is not totally disconnected. An - factor is an - regular spanning subgraph of  and  is -factorable if there are edge-disjoint -factors  such that . We shall refer as an -factorization of a graph . In this research we consider -factorization of complete graph. A graph with  vertices is called a complete graph if every pair of distinct vertices is joined by an edge and it is denoted by . We look into the possibility of factorizing  with added limitations coming in relation to the rows of generalized Hadamard matrix over a cyclic group. Over a cyclic group  of prime order , a square matrix  of order  all of whose elements are the  root of unity is called a generalized Hadamard matrix if , where  is the conjugate transpose of matrix  and  is the identity matrix of order . In the present work, generalized Hadamard matrices over a cyclic group  have been considered. We prove that the factorization is possible for  in the case of the limitation 1, namely, If an edge  belongs to the factor , then the and  entries of the corresponding generalized Hadamard matrix should be different in the   row. In Particular,  number of rows in the generalized Hadamard matrices is used to form -factorization of complete graphs. We discuss some illustrative examples that might be used for studying the factorization of complete graphs.

Sign in / Sign up

Export Citation Format

Share Document