scholarly journals Another Approach to Non-Repetitive Colorings of Graphs of Bounded Degree

10.37236/9667 ◽  
2020 ◽  
Vol 27 (3) ◽  
Author(s):  
Matthieu Rosenfeld
Keyword(s):  
Chromatic Number ◽  
Upper Bound ◽  
Upper Bounds ◽  
Maximal Degree ◽  
Chromatic Numbers ◽  
Compression Method ◽  
Bounded Degree ◽  
A Minor ◽  
Entropy Compression ◽  
Minor Improvement

We propose a new proof technique that applies to the same problems as the  Lovász Local Lemma or the entropy-compression method. We present this approach in the context of non-repetitive colorings and we use it to improve upper-bounds relating different non-repetitive chromatic numbers to the maximal degree of a graph. It seems that there should be other interesting applications of the presented approach. In terms of upper-bounds our approach seems to be as strong as entropy-compression, but the proofs are more elementary and shorter. The applications we provide in this paper are upper bounds for graphs of maximal degree at most $\Delta$: a minor improvement on the upper-bound of the non-repetitive chromatic number, a $4.25\Delta +o(\Delta)$ upper-bound on the weak total non-repetitive chromatic number, and a $ \Delta^2+\frac{3}{2^{1/3}}\Delta^{5/3}+ o(\Delta^{5/3})$ upper-bound on the total non-repetitive chromatic number of graphs. This last result implies the same upper-bound for the non-repetitive chromatic index of graphs, which improves the best known bound. 

Conflict-Free Colourings of Uniform Hypergraphs With Few Edges

2012 ◽  
Vol 21 (4) ◽  
pp. 611-622 ◽  
Author(s):  
A. KOSTOCHKA ◽  
M. KUMBHAT ◽  
T. ŁUCZAK
Keyword(s):  
Chromatic Number ◽  
Upper Bound ◽  
Upper Bounds ◽  
Uniform Hypergraph ◽  
Lower And Upper Bounds ◽  
Uniform Hypergraphs ◽  
Minimum Number

A colouring of the vertices of a hypergraph is called conflict-free if each edge e of contains a vertex whose colour does not repeat in e. The smallest number of colours required for such a colouring is called the conflict-free chromatic number of , and is denoted by χCF(). Pach and Tardos proved that for an (2r − 1)-uniform hypergraph with m edges, χCF() is at most of the order of rm1/r log m, for fixed r and large m. They also raised the question whether a similar upper bound holds for r-uniform hypergraphs. In this paper we show that this is not necessarily the case. Furthermore, we provide lower and upper bounds on the minimum number of edges of an r-uniform simple hypergraph that is not conflict-free k-colourable.

scholarly journals Upper bounds on the non-3-colourability threshold of random graphs

2002 ◽  
Vol Vol. 5 ◽  
Author(s):  
Nikolaos Fountoulakis ◽  
Colin McDiarmid
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Upper Bounds ◽  
Average Degree ◽  
Expected Number ◽  
International Audience ◽  
Full Analysis ◽  
A Minor ◽  
Minor Improvement

International audience We present a full analysis of the expected number of 'rigid' 3-colourings of a sparse random graph. This shows that, if the average degree is at least 4.99, then as n → ∞ the expected number of such colourings tends to 0 and so the probability that the graph is 3-colourable tends to 0. (This result is tight, in that with average degree 4.989 the expected number tends to ∞.) This bound appears independently in Kaporis \textitet al. [Kap]. We then give a minor improvement, showing that the probability that the graph is 3-colourable tends to 0 if the average degree is at least 4.989.

Graph coloring approach with new upper bounds for the chromatic number: team building application

2018 ◽  
Vol 52 (3) ◽  
pp. 807-818
Author(s):  
Assia Gueham ◽  
Anass Nagih ◽  
Hacene Ait Haddadene ◽  
Malek Masmoudi
Keyword(s):  
Chromatic Number ◽  
Graph Coloring ◽  
Upper Bound ◽  
Team Building ◽  
Upper Bounds ◽  
Real Case ◽  
Soft Constraints ◽  
Empirical Comparison ◽  
Hard And Soft Constraints

In this paper, we focus on the coloration approach and estimation of chromatic number. First, we propose an upper bound of the chromatic number based on the orientation algorithm described in previous studies. This upper bound is further improved by developing a novel coloration algorithm. Second, we make a theoretical and empirical comparison of our bounds with Brooks’s bound and Reed’s conjecture for class of triangle-free graphs. Third, we propose an adaptation of our algorithm to deal with the team building problem respecting several hard and soft constraints. Finally, a real case study from healthcare domain is considered for illustration.

scholarly journals Improved Bounds for Centered Colorings

2021 ◽  
Author(s):  
Michał Dębski ◽  
Piotr Micek ◽  
Felix Schröder ◽  
Stefan Felsner
Keyword(s):  
Upper Bound ◽  
Planar Graphs ◽  
Structure Theorem ◽  
Upper Bounds ◽  
Log P ◽  
Connected Subgraph ◽  
Bounded Degree ◽  
Graph Classes ◽  
Proof Techniques ◽  
Bounded Degree Graphs

A vertex coloring $\phi$ of a graph $G$ is $p$-centered if for every connected subgraph $H$ of $G$ either $\phi$ uses more than $p$ colors on $H$ or there is a color that appears exactly once on $H$. Centered colorings form one of the families of parameters that allow to capture notions of sparsity of graphs: A class of graphs has bounded expansion if and only if there is a function $f$ such that for every $p\geq1$, every graph in the class admits a $p$-centered coloring using at most $f(p)$ colors. In this paper, we give upper bounds for the maximum number of colors needed in a $p$-centered coloring of graphs from several widely studied graph classes. We show that: (1) planar graphs admit $p$-centered colorings with $O(p^3\log p)$ colors where the previous bound was $O(p^{19})$; (2) bounded degree graphs admit $p$-centered colorings with $O(p)$ colors while it was conjectured that they may require exponential number of colors. All these upper bounds imply polynomial algorithms for computing the colorings. Prior to this work there were no non-trivial lower bounds known. We show that: (4) there are graphs of treewidth $t$ that require $\binom{p+t}{t}$ colors in any $p$-centered coloring. This bound matches the upper bound; (5) there are planar graphs that require $\Omega(p^2\log p)$ colors in any $p$-centered coloring. We also give asymptotically tight bounds for outerplanar graphs and planar graphs of treewidth $3$. We prove our results with various proof techniques. The upper bound for planar graphs involves an application of a recent structure theorem while the upper bound for bounded degree graphs comes from the entropy compression method. We lift the result for bounded degree graphs to graphs avoiding a fixed topological minor using the Grohe-Marx structure theorem.

Disjointness graphs of segments in the space

2020 ◽  
pp. 1-15
Author(s):  
János Pach ◽  
Gábor Tardos ◽  
Géza Tóth
Keyword(s):  
Chromatic Number ◽  
Pairwise Disjoint ◽  
Clique Number ◽  
Upper Bounds ◽  
Efficient Algorithms ◽  
Np Hard ◽  
Chromatic Numbers ◽  
Lines In Space ◽  
Vertex Set

Abstract The disjointness graph G = G(𝒮) of a set of segments 𝒮 in ${\mathbb{R}^d}$ , $$d \ge 2$$ , is a graph whose vertex set is 𝒮 and two vertices are connected by an edge if and only if the corresponding segments are disjoint. We prove that the chromatic number of G satisfies $\chi (G) \le {(\omega (G))^4} + {(\omega (G))^3}$ , where ω(G) denotes the clique number of G. It follows that 𝒮 has Ω(n1/5) pairwise intersecting or pairwise disjoint elements. Stronger bounds are established for lines in space, instead of segments. We show that computing ω(G) and χ(G) for disjointness graphs of lines in space are NP-hard tasks. However, we can design efficient algorithms to compute proper colourings of G in which the number of colours satisfies the above upper bounds. One cannot expect similar results for sets of continuous arcs, instead of segments, even in the plane. We construct families of arcs whose disjointness graphs are triangle-free (ω(G) = 2), but whose chromatic numbers are arbitrarily large.

scholarly journals Exact Distance Colouring in Trees

2018 ◽  
Vol 28 (2) ◽  
pp. 177-186 ◽  
Author(s):  
NICOLAS BOUSQUET ◽  
LOUIS ESPERET ◽  
ARARAT HARUTYUNYAN ◽  
RÉMI DE JOANNIS DE VERCLOS
Keyword(s):  
Planar Graph ◽  
Chromatic Number ◽  
Upper Bound ◽  
Short Note ◽  
Clique Number ◽  
Negative Answer ◽  
Bounded Degree ◽  
Study Interval ◽  
Simple Corollary ◽  
Bounded Degree Trees

For an integer q ⩾ 2 and an even integer d, consider the graph obtained from a large complete q-ary tree by connecting with an edge any two vertices at distance exactly d in the tree. This graph has clique number q + 1, and the purpose of this short note is to prove that its chromatic number is Θ((d log q)/log d). It was not known that the chromatic number of this graph grows with d. As a simple corollary of our result, we give a negative answer to a problem of van den Heuvel and Naserasr, asking whether there is a constant C such that for any odd integer d, any planar graph can be coloured with at most C colours such that any pair of vertices at distance exactly d have distinct colours. Finally, we study interval colouring of trees (where vertices at distance at least d and at most cd, for some real c > 1, must be assigned distinct colours), giving a sharp upper bound in the case of bounded degree trees.

scholarly journals Locating-chromatic number of the edge-amalgamation of trees

2020 ◽  
Vol 4 (2) ◽  
pp. 126
Author(s):  
Dian Kastika Syofyan ◽  
Edy Tri Baskoro ◽  
Hilda Assiyatun
Keyword(s):  
Chromatic Number ◽  
Upper Bounds ◽  
Title Page ◽  
Metric Dimension ◽  
Chromatic Numbers ◽  
Lower And Upper Bounds ◽  
Vertex Set ◽  
Partition Dimension ◽  
Special Case

<div class="page" title="Page 1"><div class="layoutArea"><div class="column"><p><span>The investigation on the locating-chromatic number of a graph was initiated by Chartrand </span><span>et al. </span><span>(2002). This concept is in fact a special case of the partition dimension of a graph. This topic has received much attention. However, the results are still far from satisfaction. We can define the locating-chromatic number of a graph </span><span>G </span><span>as the smallest integer </span><span>k </span><span>such that there exists a </span><span>k</span><span>-partition of the vertex-set of </span><span>G </span><span>such that all vertices have distinct coordinates with respect to this partition. As we know that the metric dimension of a tree is completely solved. However, the locating-chromatic numbers for most of trees are still open. For </span><span><em>i</em> </span><span>= 1</span><span>, </span><span>2</span><span>, . . . , <em>t</em>, </span><span>let </span><em>T</em><span>i </span><span>be a tree with a fixed edge </span><span>e</span><span>o</span><span>i </span><span>called the terminal edge. The edge-amalgamation of all </span><span>T</span><span>i</span><span>s </span><span>denoted by Edge-Amal</span><span>{</span><span>T</span><span>i</span><span>;</span><span>e</span><span>o</span><span>i</span><span>} </span><span>is a tree formed by taking all the </span><span>T</span><span>i</span><span>s and identifying their terminal edges. In this paper, we study the locating-chromatic number of the edge-amalgamation of arbitrary trees. We give lower and upper bounds for their locating-chromatic numbers and show that the bounds are tight.</span></p></div></div></div>

A Unique Approach on Upper Bounds for the Chromatic Number of Total Graphs

2012 ◽  
Vol 5 (4) ◽  
pp. 240-246
Author(s):  
J. Venkateswara Rao ◽  
R.V.N. Srinivasa Rao
Keyword(s):  
Chromatic Number ◽  
Upper Bounds ◽  
Unique Approach

Psychiatric illness, medication and driving: an audit of documentation

2015 ◽  
Vol 33 (3) ◽  
pp. 171-174 ◽  
Author(s):  
A. Gallagher ◽  
S. Shah ◽  
W. Abassi ◽  
E. Walsh
Keyword(s):  
Road Safety ◽  
Psychiatric Patients ◽  
Fitness To Drive ◽  
Team Members ◽  
The Road ◽  
Significant Difference ◽  
New Guidelines ◽  
A Minor ◽  
Inpatient Psychiatric Unit ◽  
Minor Improvement

ObjectivesGuidelines on advising patients on fitness to drive have been published recently by the Road Safety Authority in collaboration with the Royal College of Physicians of Ireland. The aim of this audit is to assess if the new guidelines are being adhered to.MethodExamination of the documentation and adherence to the guidelines in the inpatient psychiatric unit, Mayo General Hospital.ResultsOf the 100 patients included in audit cycle one, none had any specific documentation about driving. One patient was admitted with alcohol misuse and was driving. On re-auditing, following presentation at academic meeting and education of team members on the guidelines, there was a minor improvement of 7%.ConclusionThere was no significant difference in documentation on re-audit. However, an increase of 7% is nonetheless encouraging. Information concerning driving should be a standard part of advice given to all psychiatric patients.

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