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 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.

scholarly journals Minimum Number of Monotone Subsequences of Length 4 in Permutations

2014 ◽  
Vol 24 (4) ◽  
pp. 658-679 ◽  
Author(s):  
JÓZSEF BALOGH ◽  
PING HU ◽  
BERNARD LIDICKÝ ◽  
OLEG PIKHURKO ◽  
BALÁZS UDVARI ◽  
...  
Keyword(s):  
Lower Bound ◽  
Complete Graphs ◽  
Formula Group ◽  
Minimum Number ◽  
Additional Stability ◽  
Edge Colourings ◽  
Image Position

We show that for every sufficiently largen, the number of monotone subsequences of length four in a permutation onnpoints is at least\begin{equation*} \binom{\lfloor{n/3}\rfloor}{4} + \binom{\lfloor{(n+1)/3}\rfloor}{4} + \binom{\lfloor{(n+2)/3}\rfloor}{4}. \end{equation*}Furthermore, we characterize all permutations on [n] that attain this lower bound. The proof uses the flag algebra framework together with some additional stability arguments. This problem is equivalent to some specific type of edge colourings of complete graphs with two colours, where the number of monochromaticK4is minimized. We show that all the extremal colourings must contain monochromaticK4only in one of the two colours. This translates back to permutations, where all the monotone subsequences of length four are all either increasing, or decreasing only.

Ramsey Numbers of Some Bipartite Graphs Versus Complete Graphs

2010 ◽  
Vol 27 (1) ◽  
pp. 121-128 ◽  
Author(s):  
Tao Jiang ◽  
Michael Salerno
Keyword(s):  
Bipartite Graphs ◽  
Complete Graphs ◽  
Ramsey Numbers

scholarly journals Constrained Ramsey Numbers

2009 ◽  
Vol 18 (1-2) ◽  
pp. 247-258 ◽  
Author(s):  
PO-SHEN LOH ◽  
BENNY SUDAKOV
Keyword(s):  
Complete Graph ◽  
Ramsey Number ◽  
Ramsey Numbers ◽  
Ramsey Theorem ◽  
Logarithmic Factor ◽  
Subgraph Isomorphic ◽  
Rate Of Growth ◽  
Special Cases ◽  
Edge Colouring

For two graphs S and T, the constrained Ramsey number f(S, T) is the minimum n such that every edge colouring of the complete graph on n vertices (with any number of colours) has a monochromatic subgraph isomorphic to S or a rainbow subgraph isomorphic to T. Here, a subgraph is said to be rainbow if all of its edges have different colours. It is an immediate consequence of the Erdős–Rado Canonical Ramsey Theorem that f(S, T) exists if and only if S is a star or T is acyclic. Much work has been done to determine the rate of growth of f(S, T) for various types of parameters. When S and T are both trees having s and t edges respectively, Jamison, Jiang and Ling showed that f(S, T) ≤ O(st2) and conjectured that it is always at most O(st). They also mentioned that one of the most interesting open special cases is when T is a path. In this paper, we study this case and show that f(S, Pt) = O(st log t), which differs only by a logarithmic factor from the conjecture. This substantially improves the previous bounds for most values of s and t.

scholarly journals On the Ramsey numbers for stars versus complete graphs

2010 ◽  
Vol 31 (7) ◽  
pp. 1680-1688 ◽  
Author(s):  
L. Boza ◽  
M. Cera ◽  
P. García-Vázquez ◽  
M.P. Revuelta
Keyword(s):  
Complete Graphs ◽  
Ramsey Numbers

Covering Two-Edge-Coloured Complete Graphs with Two Disjoint Monochromatic Cycles

2008 ◽  
Vol 17 (4) ◽  
pp. 471-486 ◽  
Author(s):  
PETER ALLEN
Keyword(s):  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Complete Graphs ◽  
Regularity Lemma ◽  
Alternative Method ◽  
Vertex Disjoint ◽  
Edge Colouring

In 1998 Łuczak Rödl and Szemerédi [7] proved, by means of the Regularity Lemma, that there exists n0 such that, for any n ≥ n0 and two-edge-colouring of Kn, there exists a pair of vertex-disjoint monochromatic cycles of opposite colours covering the vertices of Kn. In this paper we make use of an alternative method of finding useful structure in a graph, leading to a proof of the same result with a much smaller value of n0. The proof gives a polynomial-time algorithm for finding the two cycles.

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.

scholarly journals An Improvement to Mathon's Cyclotomic Ramsey Colorings

10.37236/239 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Xiaodong Xu ◽  
Stanisław P. Radziszowski
Keyword(s):  
Lower Bound ◽  
Complete Graphs ◽  
Maximum Order ◽  
Ramsey Numbers

In this note we show how to extend Mathon's cyclotomic colorings of the edges of some complete graphs without increasing the maximum order of monochromatic complete subgraphs. This improves the well known lower bound construction for multicolor Ramsey numbers, in particular we obtain $R_3(7) \ge 3214$.

Sign in / Sign up

Export Citation Format

Share Document