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.

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.

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 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 On the Oriented Game Chromatic Number

10.37236/1613 ◽  
2000 ◽  
Vol 8 (2) ◽  
Author(s):  
J. Nešetřil ◽  
E. Sopena
Keyword(s):  
Chromatic Number ◽  
Outerplanar Graph ◽  
Game Chromatic Number ◽  
Oriented Path

We consider the oriented version of a coloring game introduced by Bodlaender [On the complexity of some coloring games, Internat. J. Found. Comput. Sci. 2 (1991), 133–147]. We prove that every oriented path has oriented game chromatic number at most 7 (and this bound is tight), that every oriented tree has oriented game chromatic number at most 19 and that there exists a constant $t$ such that every oriented outerplanar graph has oriented game chromatic number at most $t$.

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 Bound Propagation

2003 ◽  
Vol 19 ◽  
pp. 139-154 ◽  
Author(s):  
M. Leisink ◽  
B. Kappen
Keyword(s):  
Linear Programming ◽  
Lower Bounds ◽  
Graphical Model ◽  
Probability Distributions ◽  
Directed Graphs ◽  
Upper And Lower Bounds ◽  
Marginal Probability ◽  
Practical Applications ◽  
Marginal Values ◽  
Small Clusters

In this article we present an algorithm to compute bounds on the marginals of a graphical model. For several small clusters of nodes upper and lower bounds on the marginal values are computed independently of the rest of the network. The range of allowed probability distributions over the surrounding nodes is restricted using earlier computed bounds. As we will show, this can be considered as a set of constraints in a linear programming problem of which the objective function is the marginal probability of the center nodes. In this way knowledge about the maginals of neighbouring clusters is passed to other clusters thereby tightening the bounds on their marginals. We show that sharp bounds can be obtained for undirected and directed graphs that are used for practical applications, but for which exact computations are infeasible.

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 Anagram-Free Graph Colouring

10.37236/6267 ◽  
2018 ◽  
Vol 25 (2) ◽  
Author(s):  
Tim E. Wilson ◽  
David R. Wood
Keyword(s):  
Chromatic Number ◽  
Lower Bounds ◽  
Maximum Degree ◽  
Empty Word ◽  
Upper And Lower Bounds ◽  
Graph Colouring ◽  
Free Graph ◽  
Random Structures ◽  
Edge Colouring

An anagram is a word of the form $WP$ where $W$ is a non-empty word and $P$ is a permutation of $W$. We study anagram-free graph colouring and give bounds on the chromatic number. Alon et al.[Random Structures & Algorithms 2002] asked whether anagram-free chromatic number is bounded by a function of the maximum degree. We answer  this question in the negative by constructing graphs with maximum degree 3 and unbounded anagram-free chromatic number. We also prove upper and lower bounds on the anagram-free chromatic number of trees in terms of their radius and pathwidth. Finally, we explore extensions to edge colouring and $k$-anagram-free colouring.

scholarly journals On the Generalized Distance Energy of Graphs

Mathematics ◽  
2019 ◽  
Vol 8 (1) ◽  
pp. 17 ◽  
Author(s):  
Abdollah Alhevaz ◽  
Maryam Baghipur ◽  
Hilal A. Ganie ◽  
Yilun Shang
Keyword(s):  
Lower Bounds ◽  
Complete Graph ◽  
Connected Graph ◽  
Distance Matrix ◽  
Wiener Index ◽  
Diagonal Matrix ◽  
Upper And Lower Bounds ◽  
Star Graph ◽  
Extremal Graphs ◽  
Generalized Distance

The generalized distance matrix D α ( G ) of a connected graph G is defined as D α ( G ) = α T r ( G ) + ( 1 − α ) D ( G ) , where 0 ≤ α ≤ 1 , D ( G ) is the distance matrix and T r ( G ) is the diagonal matrix of the node transmissions. In this paper, we extend the concept of energy to the generalized distance matrix and define the generalized distance energy E D α ( G ) . Some new upper and lower bounds for the generalized distance energy E D α ( G ) of G are established based on parameters including the Wiener index W ( G ) and the transmission degrees. Extremal graphs attaining these bounds are identified. It is found that the complete graph has the minimum generalized distance energy among all connected graphs, while the minimum is attained by the star graph among trees of order n.

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