scholarly journals A rainbow blow-up lemma for almost optimally bounded edge-colourings

2020 ◽  
Vol 8 ◽  
Author(s):  
Stefan Ehard ◽  
Stefan Glock ◽  
Felix Joos
Keyword(s):  
Blow Up ◽  
Boundedness Condition ◽  
Bounded Degree ◽  
Graph Decompositions ◽  
Spanning Subgraph ◽  
Double Covers ◽  
Edge Colourings ◽  
Host Graph ◽  
Edge Colouring

Abstract A subgraph of an edge-coloured graph is called rainbow if all its edges have different colours. We prove a rainbow version of the blow-up lemma of Komlós, Sárközy, and Szemerédi that applies to almost optimally bounded colourings. A corollary of this is that there exists a rainbow copy of any bounded-degree spanning subgraph H in a quasirandom host graph G, assuming that the edge-colouring of G fulfills a boundedness condition that is asymptotically best possible. This has many applications beyond rainbow colourings: for example, to graph decompositions, orthogonal double covers, and graph labellings.

scholarly journals Degree Ramsey Numbers of Graphs

2012 ◽  
Vol 21 (1-2) ◽  
pp. 229-253 ◽  
Author(s):  
WILLIAM B. KINNERSLEY ◽  
KEVIN G. MILANS ◽  
DOUGLAS B. WEST
Keyword(s):  
Maximum Degree ◽  
Double Star ◽  
Ramsey Number ◽  
Bounded Degree ◽  
Ramsey Numbers ◽  
Colour Class ◽  
Bounded Degree Graphs ◽  
Edge Colourings ◽  
Edge Colouring ◽  
Monochromatic Copy

Let HG mean that every s-colouring of E(H) produces a monochromatic copy of G in some colour class. Let the s-colour degree Ramsey number of a graph G, written RΔ(G; s), be min{Δ(H): HG}. If T is a tree in which one vertex has degree at most k and all others have degree at most ⌈k/2⌉, then RΔ(T; s) = s(k − 1) + ϵ, where ϵ = 1 when k is odd and ϵ = 0 when k is even. For general trees, RΔ(T; s) ≤ 2s(Δ(T) − 1).To study sharpness of the upper bound, consider the double-starSa,b, the tree whose two non-leaf vertices have degrees a and b. If a ≤ b, then RΔ(Sa,b; 2) is 2b − 2 when a < b and b is even; it is 2b − 1 otherwise. If s is fixed and at least 3, then RΔ(Sb,b;s) = f(s)(b − 1) − o(b), where f(s) = 2s − 3.5 − O(s−1).We prove several results about edge-colourings of bounded-degree graphs that are related to degree Ramsey numbers of paths. Finally, for cycles we show that RΔ(C2k + 1; s) ≥ 2s + 1, that RΔ(C2k; s) ≥ 2s, and that RΔ(C4;2) = 5. For the latter we prove the stronger statement that every graph with maximum degree at most 4 has a 2-edge-colouring such that the subgraph in each colour class has girth at least 5.

Local Anti-Ramsey Numbers of Graphs

2003 ◽  
Vol 12 (5-6) ◽  
pp. 495-511 ◽  
Author(s):  
Maria Axenovich ◽  
Tao Jiang ◽  
Z Tuza
Keyword(s):  
Complete Graphs ◽  
Ramsey Numbers ◽  
Edge Colourings ◽  
Edge Colouring

A subgraph H in an edge-colouring is properly coloured if incident edges of H are assigned different colours, and H is rainbow if no two edges of H are assigned the same colour. We study properly coloured subgraphs and rainbow subgraphs forced in edge-colourings of complete graphs in which each vertex is incident to a large number of colours.

scholarly journals A Note on Edge-Colourings Avoiding Rainbow $K_4$ and Monochromatic $K_m$

10.37236/257 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Veselin Jungić ◽  
Tomáš Kaiser ◽  
Daniel Král'
Keyword(s):  
Lower Bound ◽  
Complete Graph ◽  
Upper Bound ◽  
Projective Planes ◽  
Ramsey Number ◽  
Complete Subgraph ◽  
Finite Projective Planes ◽  
Edge Colourings ◽  
Edge Colouring

We study the mixed Ramsey number $maxR(n,{K_m},{K_r})$, defined as the maximum number of colours in an edge-colouring of the complete graph $K_n$, such that $K_n$ has no monochromatic complete subgraph on $m$ vertices and no rainbow complete subgraph on $r$ vertices. Improving an upper bound of Axenovich and Iverson, we show that $maxR(n,{K_m},{K_4}) \leq n^{3/2}\sqrt{2m}$ for all $m\geq 3$. Further, we discuss a possible way to improve their lower bound on $maxR(n,{K_4},{K_4})$ based on incidence graphs of finite projective planes.

scholarly journals Neighbour$\,$–$\,$Distinguishing Edge Colourings of Random Regular Graphs

10.37236/1103 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Catherine Greenhill ◽  
Andrzej Ruciński
Keyword(s):  
Regular Graph ◽  
Regular Graphs ◽  
Edge Colourings ◽  
Almost All ◽  
Edge Colouring

A proper edge colouring of a graph is neighbour-distinguishing if for all pairs of adjacent vertices $v$, $w$ the set of colours appearing on the edges incident with $v$ is not equal to the set of colours appearing on the edges incident with $w$. Let ${\rm ndi}(G)$ be the least number of colours required for a proper neighbour-distinguishing edge colouring of $G$. We prove that for $d\geq 4$, a random $d$-regular graph $G$ on $n$ vertices asymptotically almost surely satisfies ${\rm ndi}(G)\leq \lceil 3d/2\rceil$. This verifies a conjecture of Zhang, Liu and Wang for almost all 4-regular graphs.

Wild edge colourings of graphs

2004 ◽  
Vol 69 (1) ◽  
pp. 255-264
Author(s):  
Mirna Džamonja ◽  
Péter Komjáth ◽  
Charles Morgan
Keyword(s):  
Supercompact Cardinal ◽  
Strong Limit ◽  
Limit Cardinal ◽  
Colour Class ◽  
Edge Colourings ◽  
Vertex Colouring ◽  
Edge Colouring

AbstractWe prove consistent, assuming there is a supercompact cardinal, that there is a singular strong limit cardinal μ, of cofinality ω, such that every μ+-chromatic graph X on μ+ has an edge colouring c of X into μ colours for which every vertex colouring g of X into at most μ many colours has a g-colour class on which c takes every value.The paper also contains some generalisations of the above statement in which μ+ is replaced by other cardinals > μ.

scholarly journals The bandwidth theorem for locally dense graphs

2020 ◽  
Vol 8 ◽  
Author(s):  
Katherine Staden ◽  
Andrew Treglown
Keyword(s):  
Maximum Degree ◽  
Blow Up ◽  
Minimum Degree ◽  
Bounded Degree ◽  
Dense Graphs ◽  
Vertex Graph ◽  
Delta G

Abstract The bandwidth theorem of Böttcher, Schacht, and Taraz [Proof of the bandwidth conjecture of Bollobás andKomlós, Mathematische Annalen, 2009] gives a condition on the minimum degree of an n-vertex graph G that ensures G contains every r-chromatic graph H on n vertices of bounded degree and of bandwidth $o(n)$ , thereby proving a conjecture of Bollobás and Komlós [The Blow-up Lemma, Combinatorics, Probability, and Computing, 1999]. In this paper, we prove a version of the bandwidth theorem for locally dense graphs. Indeed, we prove that every locally dense n-vertex graph G with $\delta (G)> (1/2+o(1))n$ contains as a subgraph any given (spanning) H with bounded maximum degree and sublinear bandwidth.

scholarly journals On Almost-Regular Edge Colourings of Hypergraphs

10.37236/5826 ◽  
2016 ◽  
Vol 23 (4) ◽  
Author(s):  
Darryn Bryant
Keyword(s):  
Uniform Hypergraphs ◽  
Edge Colourings ◽  
Edge Colouring

We prove that if ${\cal{H}}=(V({\cal{H}}),{\cal{E}}({\cal{H}}))$ is a hypergraph, $\gamma$ is an edge colouring of ${\cal{H}}$, and $S\subseteq V({\cal{H}})$ such that any permutation of $S$ is an automorphism of ${\cal{H}}$, then there exists a permutation $\pi$ of ${\cal{E}}({\cal{H}})$ such that $|\pi(E)|=|E|$ and $\pi(E)\setminus S=E\setminus S$ for each $E\in{\cal{H}}({\cal{H}})$, and such that the edge colouring $\gamma'$ of ${\cal{H}}$ given by $\gamma'(E)=\gamma(\pi^{-1}(E))$ for each $E\in{\cal{E}}({\cal{H}})$ is almost regular on $S$. The proof is short and elementary. We show that a number of known results, such as Baranyai's Theorem on almost-regular edge colourings of complete $k$-uniform hypergraphs, are easy corollaries of this theorem.

scholarly journals Total Graph Interpretation of the Numbers of the Fibonacci Type

2015 ◽  
Vol 2015 ◽  
pp. 1-7 ◽  
Author(s):  
Urszula Bednarz ◽  
Iwona Włoch ◽  
Małgorzata Wołowiec-Musiał
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Total Graph ◽  
Graph Interpretation ◽  
Edge Colourings ◽  
Almost All ◽  
Edge Colouring

We give a total graph interpretation of the numbers of the Fibonacci type. This graph interpretation relates to an edge colouring by monochromatic paths in graphs. We will show that it works for almost all numbers of the Fibonacci type. Moreover, we give the lower bound and the upper bound for the number of all(A1,2A1)-edge colourings in trees.

Hypergraph Packing and Graph Embedding

1999 ◽  
Vol 8 (4) ◽  
pp. 363-376 ◽  
Author(s):  
V. RÖDL ◽  
A. RUCIŃSKI ◽  
A. TARAZ
Keyword(s):  
Blow Up ◽  
Sufficient Conditions ◽  
Graph Embedding ◽  
The Other ◽  
Alternative Proof ◽  
Bounded Degree ◽  
Asymptotic Case ◽  
Bipartite Case

We provide sufficient conditions for packing two hypergraphs. The emphasis is on the asymptotic case when one of the hypergraphs has a bounded degree and the other is dense. As an application, we give an alternative proof for the bipartite case of the recently developed Blow-up Lemma [12].

Positive solutions for a quasilinear system Via blow up

1993 ◽  
Vol 18 (12) ◽  
pp. 2071-2106
Author(s):  
Philippe Clément ◽  
Raúl Manásevich ◽  
Enzo Mitidieri
Keyword(s):  
Positive Solutions ◽  
Blow Up ◽  
Quasilinear System
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