scholarly journals Upper density of monochromatic infinite paths

2019 ◽  
Author(s):  
Jan Corsten ◽  
Louis DeBiasio ◽  
Ander Lamaison ◽  
Richard Lang
Keyword(s):  
Complete Graph ◽  
Edge Coloring ◽  
Classical Case ◽  
Infinite Graphs ◽  
Edge Colorings ◽  
Vertex Set ◽  
Upper Density ◽  
Countably Infinite ◽  
Positive Integers ◽  
Red Path

Ramsey Theory investigates the existence of large monochromatic substructures. Unlike the most classical case of monochromatic complete subgraphs, the maximum guaranteed length of a monochromatic path in a two-edge-colored complete graph is well-understood. Gerencsér and Gyárfás in 1967 showed that any two-edge-coloring of a complete graph Kn contains a monochromatic path with ⌊2n/3⌋+1 vertices. The following two-edge-coloring shows that this is the best possible: partition the vertices of Kn into two sets A and B such that |A|=⌊n/3⌋ and |B|=⌈2n/3⌉, and color the edges between A and B red and edges inside each of the sets blue. The longest red path has 2|A|+1 vertices and the longest blue path has |B| vertices. The main result of this paper concerns the corresponding problem for countably infinite graphs. To measure the size of a monochromatic subgraph, we associate the vertices with positive integers and consider the lower and the upper density of the vertex set of a monochromatic subgraph. The upper density of a subset A of positive integers is the limit superior of |A∩{1,...,}|/n, and the lower density is the limit inferior. The following example shows that there need not exist a monochromatic path with positive upper density such that its vertices form an increasing sequence: an edge joining vertices i and j is colored red if ⌊log2i⌋≠⌊log2j⌋, and blue otherwise. In particular, the coloring yields blue cliques with 1, 2, 4, 8, etc., vertices mutually joined by red edges. Likewise, there are constructions of two-edge-colorings such that the lower density of every monochromatic path is zero. A result of Rado from the 1970's asserts that the vertices of any k-edge-colored countably infinite complete graph can be covered by k monochromatic paths. For a two-edge-colored complete graph on the positive integers, this implies the existence of a monochromatic path with upper density at least 1/2. In 1993, Erdős and Galvin raised the problem of determining the largest c such that every two-edge-coloring of the complete graph on the positive integers contains a monochromatic path with upper density at least c. The authors solve this 25-year-old problem by showing that c=(12+8–√)/17≈0.87226.

scholarly journals On the Turán Properties of Infinite Graphs

10.37236/771 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Andrzej Dudek ◽  
Vojtěch Rödl
Keyword(s):  
Infinite Graph ◽  
Infinite Graphs ◽  
The Other ◽  
Other Hand ◽  
Turan's Theorem ◽  
Vertex Set ◽  
Turán’S Theorem ◽  
Positive Integers

Let $G^{(\infty)}$ be an infinite graph with the vertex set corresponding to the set of positive integers ${\Bbb N}$. Denote by $G^{(l)}$ a subgraph of $G^{(\infty)}$ which is spanned by the vertices $\{1,\dots,l\}$. As a possible extension of Turán's theorem to infinite graphs, in this paper we will examine how large $\liminf_{l\rightarrow \infty} {|E(G^{(l)})|\over l^2}$ can be for an infinite graph $G^{(\infty)}$, which does not contain an increasing path $I_k$ with $k+1$ vertices. We will show that for sufficiently large $k$ there are $I_k$–free infinite graphs with ${1\over 4}+{1\over 200} < \liminf_{l\rightarrow \infty} {|E(G^{(l)})|\over l^2}$. This disproves a conjecture of J. Czipszer, P. Erdős and A. Hajnal. On the other hand, we will show that $\liminf_{l\rightarrow \infty} {|E(G^{(l)})|\over l^2}\le{1\over 3}$ for any $k$ and such $G^{(\infty)}$.

scholarly journals Chromatic Graphs, Ramsey Numbers and the Flexible Atom Conjecture

10.37236/773 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Jeremy F. Alm ◽  
Roger D. Maddux ◽  
Jacob Manske
Keyword(s):  
Complete Graph ◽  
Ramsey Theory ◽  
Projective Planes ◽  
Relation Algebras ◽  
Ramsey Numbers ◽  
Edge Colorings ◽  
Finite Set ◽  
Vertex Set ◽  
Special Case

Let $K_{N}$ denote the complete graph on $N$ vertices with vertex set $V = V(K_{N})$ and edge set $E = E(K_{N})$. For $x,y \in V$, let $xy$ denote the edge between the two vertices $x$ and $y$. Let $L$ be any finite set and ${\cal M} \subseteq L^{3}$. Let $c : E \rightarrow L$. Let $[n]$ denote the integer set $\{1, 2, \ldots, n\}$. For $x,y,z \in V$, let $c(xyz)$ denote the ordered triple $\big(c(xy)$, $c(yz), c(xz)\big)$. We say that $c$ is good with respect to ${\cal M}$ if the following conditions obtain: 1. $\forall x,y \in V$ and $\forall (c(xy),j,k) \in {\cal M}$, $\exists z \in V$ such that $c(xyz) = (c(xy),j,k)$; 2. $\forall x,y,z \in V$, $c(xyz) \in {\cal M}$; and 3. $\forall x \in V \ \forall \ell\in L \ \exists \, y\in V$ such that $ c(xy)=\ell $. We investigate particular subsets ${\cal M}\subseteq L^{3}$ and those edge colorings of $K_{N}$ which are good with respect to these subsets ${\cal M}$. We also remark on the connections of these subsets and colorings to projective planes, Ramsey theory, and representations of relation algebras. In particular, we prove a special case of the flexible atom conjecture.

scholarly journals An Explicit Edge-Coloring of $K_n$ with Six Colors on Every $K_5$

10.37236/7852 ◽  
2019 ◽  
Vol 26 (4) ◽  
Author(s):  
Alex Cameron
Keyword(s):  
Lower Bound ◽  
Complete Graph ◽  
Upper Bound ◽  
Edge Coloring ◽  
Minimum Number ◽  
Positive Integers

Let $p$ and $q$ be positive integers such that $1 \leq q \leq {p \choose 2}$. A $(p,q)$-coloring of the complete graph on $n$ vertices $K_n$ is an edge coloring for which every $p$-clique contains edges of at least $q$ distinct colors. We denote the minimum number of colors needed for such a $(p,q)$-coloring of $K_n$ by $f(n,p,q)$. This is known as the Erdös-Gyárfás function. In this paper we give an explicit $(5,6)$-coloring with $n^{1/2+o(1)}$ colors. This improves the best known upper bound of $f(n,5,6)=O\left(n^{3/5}\right)$ given by Erdös and Gyárfás, and comes close to matching the order of the best known lower bound, $f(n,5,6) = \Omega\left(n^{1/2}\right)$.

scholarly journals Restricted Extension of Sparse Partial Edge Colorings of Complete graphs

10.37236/9552 ◽  
2021 ◽  
Vol 28 (2) ◽  
Author(s):  
Carl Johan Casselgren ◽  
Lan Anh Pham
Keyword(s):  
Complete Graph ◽  
Edge Coloring ◽  
Positive Answer ◽  
Complete Graphs ◽  
Edge Colorings ◽  
Proper Edge Coloring

Given a partial edge coloring of a complete graph $K_n$ and lists of allowed colors for the non-colored edges of $K_n$, can we extend the partial edge coloring to a proper edge coloring of $K_n$ using only colors from the lists? We prove that this question has a positive answer in the case when both the partial edge coloring and the color lists satisfy certain sparsity conditions.

scholarly journals On Colorful Edge Triples in Edge-Colored Complete Graphs

2020 ◽  
Vol 36 (6) ◽  
pp. 1623-1637
Author(s):  
Gábor Simonyi
Keyword(s):  
Complete Graph ◽  
Edge Coloring ◽  
Complete Graphs ◽  
Edge Colorings ◽  
Minimum Number ◽  
Ramsey Type

Abstract An edge-coloring of the complete graph $$K_n$$ K n we call F-caring if it leaves no F-subgraph of $$K_n$$ K n monochromatic and at the same time every subset of |V(F)| vertices contains in it at least one completely multicolored version of F. For the first two meaningful cases, when $$F=K_{1,3}$$ F = K 1 , 3 and $$F=P_4$$ F = P 4 we determine for infinitely many n the minimum number of colors needed for an F-caring edge-coloring of $$K_n$$ K n . An explicit family of $$2\lceil \log _2 n\rceil $$ 2 ⌈ log 2 n ⌉ 3-edge-colorings of $$K_n$$ K n so that every quadruple of its vertices contains a totally multicolored $$P_4$$ P 4 in at least one of them is also presented. Investigating related Ramsey-type problems we also show that the Shannon (OR-)capacity of the Grötzsch graph is strictly larger than that of the cycle of length 5.

scholarly journals Rainbow $H$-Factors of Complete $s$-Uniform $r$-Partite Hypergraphs

10.37236/901 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Ailian Chen ◽  
Fuji Zhang ◽  
Hao Li
Keyword(s):  
Chromatic Number ◽  
Edge Coloring ◽  
Uniform Hypergraph ◽  
Vertex Partition ◽  
Proper Edge Coloring ◽  
Vertex Set ◽  
Positive Integers

We say a $s$-uniform $r$-partite hypergraph is complete, if it has a vertex partition $\{V_1,V_2,...,V_r\}$ of $r$ classes and its hyperedge set consists of all the $s$-subsets of its vertex set which have at most one vertex in each vertex class. We denote the complete $s$-uniform $r$-partite hypergraph with $k$ vertices in each vertex class by ${\cal T}_{s,r}(k)$. In this paper we prove that if $h,\ r$ and $s$ are positive integers with $2\leq s\leq r\leq h$ then there exists a constant $k=k(h,r,s)$ so that if $H$ is an $s$-uniform hypergraph with $h$ vertices and chromatic number $\chi(H)=r$ then any proper edge coloring of ${\cal T}_{s,r}(k)$ has a rainbow $H$-factor.

scholarly journals Density of Monochromatic Infinite Paths

10.37236/7758 ◽  
2018 ◽  
Vol 25 (4) ◽  
Author(s):  
Allan Lo ◽  
Nicolás Sanhueza-Matamala ◽  
Guanghui Wang
Keyword(s):  
Complete Graph ◽  
Infinite Path ◽  
Vertex Set ◽  
Upper Density ◽  
Edge Colouring

For any subset $A \subseteq \mathbb{N}$, we define its upper density to be $\limsup_{ n \rightarrow \infty } |A \cap \{ 1, \dotsc, n \}| / n$. We prove that every $2$-edge-colouring of the complete graph on $\mathbb{N}$ contains a monochromatic infinite path, whose vertex set has upper density at least $(9 + \sqrt{17})/16 \approx 0.82019$. This improves on results of Erdős and Galvin, and of DeBiasio and McKenney.

scholarly journals A note on compact and compact circular edge-colorings of graphs

2008 ◽  
Vol Vol. 10 no. 3 (Graph and Algorithms) ◽  
Author(s):  
Dariusz Dereniowski ◽  
Adam Nadolski
Keyword(s):  
Approximation Algorithm ◽  
Decision Problem ◽  
Edge Coloring ◽  
Exact Algorithm ◽  
Weighted Graphs ◽  
Edge Colorings ◽  
Odd Cycle ◽  
International Audience ◽  
Np Complete ◽  
Odd Cycles

Graphs and Algorithms International audience We study two variants of edge-coloring of edge-weighted graphs, namely compact edge-coloring and circular compact edge-coloring. First, we discuss relations between these two coloring models. We prove that every outerplanar bipartite graph admits a compact edge-coloring and that the decision problem of the existence of compact circular edge-coloring is NP-complete in general. Then we provide a polynomial time 1:5-approximation algorithm and pseudo-polynomial exact algorithm for compact circular coloring of odd cycles and prove that it is NP-hard to optimally color these graphs. Finally, we prove that if a path P2 is joined by an edge to an odd cycle then the problem of the existence of a compact circular coloring becomes NP-complete.

scholarly journals On the Chvátal-Erdős Triangle Game

10.37236/559 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
József Balogh ◽  
Wojciech Samotij
Keyword(s):  
Complete Graph ◽  
Winning Strategy ◽  
Seminal Paper ◽  
Positional Games ◽  
Positive Integers

Given a graph $G$ and positive integers $n$ and $q$, let ${\bf G}(G;n,q)$ be the game played on the edges of the complete graph $K_n$ in which the two players, Maker and Breaker, alternately claim $1$ and $q$ edges, respectively. Maker's goal is to occupy all edges in some copy of $G$; Breaker tries to prevent it. In their seminal paper on positional games, Chvátal and Erdős proved that in the game ${\bf G}(K_3;n,q)$, Maker has a winning strategy if $q < \sqrt{2n+2}-5/2$, and if $q \geq 2\sqrt{n}$, then Breaker has a winning strategy. In this note, we improve the latter of these bounds by describing a randomized strategy that allows Breaker to win the game ${\bf G}(K_3;n,q)$ whenever $q \geq (2-1/24)\sqrt{n}$. Moreover, we provide additional evidence supporting the belief that this bound can be further improved to $(\sqrt{2}+o(1))\sqrt{n}$.

scholarly journals PELABELAN TOTAL TITIK AJAIB PADA GRAF PETERSEN

2017 ◽  
Vol 1 (1) ◽  
pp. 44
Author(s):  
Chusnul Noeriansyah Poetri
Keyword(s):  
Mapping Functions ◽  
Vertex Set ◽  
Positive Integers

Suppose a graph G with vertex set V(G) and the edge set E(G) where each vertex V(G) and edge E(G) is given a one - one function and on the mapping functions using positive integers {1,2, … ,

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