scholarly journals Coloring Chains for Compression with Uncertain Priors

10.37236/6468 ◽  
2018 ◽  
Vol 25 (4) ◽  
Author(s):  
Noah Golowich
Keyword(s):  
Chromatic Number ◽  
Lower Bounds ◽  
Graph Coloring ◽  
Upper And Lower Bounds ◽  
Nice Property ◽  
Chromatic Numbers ◽  
Compression Scheme ◽  
Graph Coloring Problem ◽  
Close Relationship ◽  
Local Graph

Haramaty and Sudan considered the problem of transmitting a message between two people, Alice and Bob, when Alice's and Bob's priors on the message are allowed to differ by at most a given factor. To find a deterministic compression scheme for this problem, they showed that it is sufficient to obtain an upper bound on the chromatic number of a graph, denoted $U(N,s,k)$ for parameters $N,s,k$, whose vertices are nested sequences of subsets and whose edges are between vertices that have similar sequences of sets. In turn, there is a close relationship between the problem of determining the chromatic number of $U(N,s,k)$ and a local graph coloring problem considered by Erdős et al. We generalize the results of Erdős et al. by finding bounds on the chromatic numbers of graphs $H$ and $G$ when there is a homomorphism $\phi :H\rightarrow G$ that satisfies a nice property. We then use these results to improve upper and lower bounds on $\chi(U(N,s,k))$. 

scholarly journals Online Graph Coloring Against a Randomized Adversary

2018 ◽  
Vol 29 (04) ◽  
pp. 551-569 ◽  
Author(s):  
Elisabet Burjons ◽  
Juraj Hromkovič ◽  
Rastislav Královič ◽  
Richard Královič ◽  
Xavier Muñoz ◽  
...  
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Online Algorithms ◽  
Graph Coloring ◽  
Competitive Ratio ◽  
Upper And Lower Bounds ◽  
Coloring Problem ◽  
Single Instance ◽  
Graph Coloring Problem

We consider an online model where an adversary constructs a set of [Formula: see text] instances [Formula: see text] instead of one single instance. The algorithm knows [Formula: see text] and the adversary will choose one instance from [Formula: see text] at random to present to the algorithm. We further focus on adversaries that construct sets of [Formula: see text]-chromatic instances. In this setting, we provide upper and lower bounds on the competitive ratio for the online graph coloring problem as a function of the parameters in this model. Both bounds are linear in [Formula: see text] and matching upper and lower bound are given for a specific set of algorithms that we call “minimalistic online algorithms”.

2-Distance chromatic number of some graph products

2020 ◽  
Vol 12 (02) ◽  
pp. 2050021
Author(s):  
Ghazale Ghazi ◽  
Freydoon Rahbarnia ◽  
Mostafa Tavakoli
Keyword(s):  
Chromatic Number ◽  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Graph Products ◽  
Lexicographic Product ◽  
Graph Product ◽  
Minimum Number

This paper studies the 2-distance chromatic number of some graph product. A coloring of [Formula: see text] is 2-distance if any two vertices at distance at most two from each other get different colors. The minimum number of colors in the 2-distance coloring of [Formula: see text] is the 2-distance chromatic number and denoted by [Formula: see text]. In this paper, we obtain some upper and lower bounds for the 2-distance chromatic number of the rooted product, generalized rooted product, hierarchical product and we determine exact value for the 2-distance chromatic number of the lexicographic product.

scholarly journals A Note on the Warmth of Random Graphs with Given Expected Degrees

2014 ◽  
Vol 2014 ◽  
pp. 1-4 ◽  
Author(s):  
Yilun Shang
Keyword(s):  
Statistical Mechanics ◽  
Chromatic Number ◽  
Random Graph ◽  
Lower Bounds ◽  
Random Graphs ◽  
Upper Bound ◽  
Degree Sequence ◽  
Graph Model ◽  
Upper And Lower Bounds ◽  
Random Graph Model

We consider the random graph modelG(w)for a given expected degree sequencew=(w1,w2,…,wn). Warmth, introduced by Brightwell and Winkler in the context of combinatorial statistical mechanics, is a graph parameter related to lower bounds of chromatic number. We present new upper and lower bounds on warmth ofG(w). In particular, the minimum expected degree turns out to be an upper bound of warmth when it tends to infinity and the maximum expected degreem=O(nα)with0<α<1/2.

Complexity Analysis and Stochastic Convergence of Some Well-known Evolutionary Operators for Solving Graph Coloring Problem

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 303 ◽  
Author(s):  
Raja Marappan ◽  
Gopalakrishnan Sethumadhavan
Keyword(s):  
Asymptotic Analysis ◽  
Chromatic Number ◽  
Graph Coloring ◽  
Sufficient Conditions ◽  
Stochastic Convergence ◽  
Coloring Problem ◽  
Mutation Operators ◽  
Graph Coloring Problem ◽  
Evolutionary Operators ◽  
Evolutionary Operator

The graph coloring problem is an NP-hard combinatorial optimization problem and can be applied to various engineering applications. The chromatic number of a graph G is defined as the minimum number of colors required to color the vertex set V(G) so that no two adjacent vertices are of the same color, and different approximations and evolutionary methods can find it. The present paper focused on the asymptotic analysis of some well-known and recent evolutionary operators for finding the chromatic number. The asymptotic analysis of different crossover and mutation operators helps in choosing the better evolutionary operator to minimize the problem search space and computational complexity. The choice of the right genetic operators facilitates an evolutionary algorithm to achieve faster convergence with lesser population size N through an adequate distribution of promising genes. The selection of an evolutionary operator plays an essential role in reducing the bounds for minimum color obtained so far for some of the benchmark graphs. This research also focuses on the necessary and sufficient conditions for the global convergence of evolutionary algorithms. The stochastic convergence of recent evolutionary operators for solving graph coloring is newly analyzed.

scholarly journals Locating Chromatic Number of Powers of Paths and Cycles

Symmetry ◽  
2019 ◽  
Vol 11 (3) ◽  
pp. 389
Author(s):  
Manal Ghanem ◽  
Hasan Al-Ezeh ◽  
Ala’a Dabbour
Keyword(s):  
Chromatic Number ◽  
Lower Bounds ◽  
Minimum Distance ◽  
Upper And Lower Bounds ◽  
Color Code ◽  
Partition Class ◽  
Distinct Color

Let c be a proper k-coloring of a graph G. Let π = { R 1 , R 2 , … , R k } be the partition of V ( G ) induced by c, where R i is the partition class receiving color i. The color code c π ( v ) of a vertex v of G is the ordered k-tuple ( d ( v , R 1 ) , d ( v , R 2 ) , … , d ( v , R k ) ) , where d ( v , R i ) is the minimum distance from v to each other vertex u ∈ R i for 1 ≤ i ≤ k . If all vertices of G have distinct color codes, then c is called a locating k-coloring of G. The locating-chromatic number of G, denoted by χ L ( G ) , is the smallest k such that G admits a locating coloring with k colors. In this paper, we give a characterization of the locating chromatic number of powers of paths. In addition, we find sharp upper and lower bounds for the locating chromatic number of powers of cycles.

scholarly journals Packing chromatic number of transformation graphs

2019 ◽  
Vol 23 (Suppl. 6) ◽  
pp. 1991-1995
Author(s):  
Derya Durgun ◽  
Busra Ozen-Dortok
Keyword(s):  
Chromatic Number ◽  
Graph Coloring ◽  
Chromatic Numbers ◽  
Np Complete ◽  
Packing Chromatic Number ◽  
General Graphs

Graph coloring is an assignment of labels called colors to elements of a graph. The packing coloring was introduced by Goddard et al. [1] in 2008 which is a kind of coloring of a graph. This problem is NP-complete for general graphs. In this paper, we consider some transformation graphs and generalized their packing chromatic numbers.

scholarly journals Colouring the Square of the Cartesian Product of Trees

2011 ◽  
Vol Vol. 13 no. 2 (Graph and Algorithms) ◽  
Author(s):  
David R. Wood
Keyword(s):  
Chromatic Number ◽  
Lower Bounds ◽  
Maximum Degree ◽  
Cartesian Product ◽  
Upper And Lower Bounds ◽  
International Audience

Graphs and Algorithms International audience We prove upper and lower bounds on the chromatic number of the square of the cartesian product of trees. The bounds are equal if each tree has even maximum degree.

scholarly journals The Guessing Number of Undirected Graphs

10.37236/679 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Demetres Christofides ◽  
Klas Markström
Keyword(s):  
Network Coding ◽  
Chromatic Number ◽  
Lower Bounds ◽  
Perfect Graphs ◽  
Upper And Lower Bounds ◽  
Undirected Graphs ◽  
Fractional Chromatic Number ◽  
Winning Probability ◽  
Optimal Bounds

Riis [Electron. J. Combin., 14(1):R44, 2007] introduced a guessing game for graphs which is equivalent to finding protocols for network coding. In this paper we prove upper and lower bounds for the winning probability of the guessing game on undirected graphs. We find optimal bounds for perfect graphs and minimally imperfect graphs, and present a conjecture relating the exact value for all graphs to the fractional chromatic number.

scholarly journals On The Locating-Chromatic Numbers of Subdivisions of Friendship Graph

2020 ◽  
Vol 26 (2) ◽  
pp. 175-184
Author(s):  
Brilly Maxel Salindeho ◽  
Hilda Assiyatun ◽  
Edy Tri Baskoro
Keyword(s):  
Chromatic Number ◽  
Lower Bounds ◽  
Connected Graph ◽  
Common Vertex ◽  
Chromatic Numbers

Let c be a k-coloring of a connected graph G and let pi={C1,C2,...,Ck} be the partition of V(G) induced by c. For every vertex v of G, let c_pi(v) be the coordinate of v relative to pi, that is c_pi(v)=(d(v,C1 ),d(v,C2 ),...,d(v,Ck )), where d(v,Ci )=min{d(v,x)|x in Ci }. If every two vertices of G have different coordinates relative to pi, then c is said to be a locating k-coloring of G. The locating-chromatic number of G, denoted by chi_L (G), is the least k such that there exists a locating k-coloring of G. In this paper, we determine the locating-chromatic numbers of some subdivisions of the friendship graph Fr_t, that is the graph obtained by joining t copies of 3-cycle with a common vertex, and we give lower bounds to the locating-chromatic numbers of few other subdivisions of Fr_t.

scholarly journals Game Colouring Directed Graphs

10.37236/283 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Daqing Yang ◽  
Xuding Zhu
Keyword(s):  
Planar Graph ◽  
Chromatic Number ◽  
Lower Bounds ◽  
Directed Graphs ◽  
Upper And Lower Bounds ◽  
Outerplanar Graph ◽  
Game Chromatic Number

In this paper, a colouring game and two versions of marking games (the weak and the strong) on digraphs are studied. We introduce the weak game chromatic number $\chi_{\rm wg}(D)$ and the weak game colouring number ${\rm wgcol}(D)$ of digraphs $D$. It is proved that if $D$ is an oriented planar graph, then $\chi_{\rm wg}(D)$ $\le {\rm wgcol}(D) \le 9$, and if $D$ is an oriented outerplanar graph, then $\chi_{\rm wg}(D)$ $\le {\rm wgcol}(D) \le 4$. Then we study the strong game colouring number ${\rm sgcol}\left( D \right)$ (which was first introduced by Andres as game colouring number) of digraphs $D$. It is proved that if $D$ is an oriented planar graph, then ${\rm sgcol}\left( D \right) \le 16$. The asymmetric versions of the colouring and marking games of digraphs are also studied. Upper and lower bounds of related parameters for various classes of digraphs are obtained.

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