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### Distance graphs and the T-coloring problem

Albert Gräf
1999 Discrete Mathematics
Exploiting an equivalence between the complete graph T-coloring problem and the distance graph clique problem, it is shown that the complete graph T-coloring problem is NP-complete in the strong sense.  ...  The T-coloring problem is, given a graph G = (V, E), a set T of nonnegative integers containing 0, and a 'span' bound s 2 0, to compute an integer coloring f of the vertices of G such that If(u) -f(w)1  ...  The basic fact that makes T-graphs so interesting for the study of the T-coloring problem is that the complete graph with n vertices has a T-coloring of span <s if and only if the corresponding T-graph  ...

### Distance-Two Colorings of Barnette Graphs [article]

Tomas Feder, Pavol Hell, Carlos Subi
2018 arXiv   pre-print
We fully describe all Goodey graphs that admit a distance-two four-coloring, and characterize the remaining type-two Barnette graphs that admit a distance-two four-coloring according to their face size  ...  A distance-two r-coloring of a graph G is an assignment of r colors to the vertices of G so that any two vertices at distance at most two have different colors.  ...  With its unique distance-two coloring add the edges b s b s+1 and b t b t+1 .  ...

### Parameterized Coloring Problems on Threshold Graphs [article]

I.Vinod Reddy
2020 arXiv   pre-print
We show that Precoloring Extension is fixed-parameter tractable (FPT) parameterized by distance to clique and Equitable Coloring is FPT parameterized by the distance to threshold graphs.  ...  We also study the List k-Coloring and show that the problem is NP-complete on split graphs and it is FPT parameterized by solution size on split graphs.  ...  We showed that (a) Precoloring Extension and Equitable Coloring are FPT parameterized by distance to threshold graphs and (b) List k-Coloring is FPT parameterized by k on split graphs.  ...

### Chromatic Coloring of Distance Graphs I

V. Yegnanarayanan, Department of Mathematics, Kalasalingam Academy for Research and Education, Deemed to be University, Krishnankoil, Srivilliputhur (Tamil Nadu), India.
2021 VOLUME-8 ISSUE-10, AUGUST 2019, REGULAR ISSUE
We prove some interesting results related to the computation of chromatic number of certain distance graphs and also discuss some open problems.  ...  The primary aim of this paper is to publicize various problems regarding chromatic coloring of finite, simple and undirected graphs.  ...  G1 is a Unit Distance Graph and G2 is a Forbidden Graph Note that K4, the complete graph on 4 vertices and K2,3, the complete bipartite graph with two vertices in one partite set and three vertices in  ...

### Distributed self-stabilizing placement of replicated resources in emerging networks

Bong-Jun Ko, D. Rubenstein
2005 IEEE/ACM Transactions on Networking
Our combination of theoretical results and simulation prove stabilization of the protocol, and evaluate its performance in the context of convergence time, message transmissions, and color distance.  ...  We describe our protocol in the context of a graph with colored nodes, where a node's color indicates the replica/task that it is assigned.  ...  Coloring Distance We evaluate the performance of our protocol by comparing the color distances in stable graphs generated by the protocol to the super-optimal distances and to distances in graphs where  ...

### Open Distance Pattern Coloring of a Graph

K. A. Germina, P. T. Marykutty
2012 Journal of Fuzzy Set Valued Analysis
Case 2: Center of T is K 2 Let x, y be the center of T. Choose x as the root vertex of T and M = {y, v} for some vertex v such that xv ∈ E(T ).  ...  Motivated from the definition of dpd (odpu)-graphs and the classic k-coloring problem, we define M -open distance pattern coloring of a graph G as follows.  ...  Since G is a unicyclic graph, no two adjacent vertices have same open distance pattern coloring and hence G is an open distance pattern colorable graph.  ...

### Solutions of some L(2,1)-coloring related open problems

Nibedita Mandal, Pratima Panigrahi
2016 Discussiones Mathematicae Graph Theory
An L(2, 1)-coloring (or labeling) of a graph G is a vertex coloring f :  ...  Acknowledgements We are thankful to the referees for their valuable comments and suggestions.  ...  Since vertices in V (T ) − V (T 1 ) are colored following greedy algorithm and since the distance between two vertices in T 1 is the same as the distance in T , the coloring obtained for T is an L(2, 1  ...

### Radiocoloring in planar graphs: Complexity and approximations

D.A. Fotakis, S.E. Nikoletseas, V.G. Papadopoulou, P.G. Spirakis
2005 Theoretical Computer Science
The FAP is usually modelled by variations of the graph coloring problem. A Radiocoloring The number of discrete frequencies and the range of frequencies used are called order and span, respectively.  ...  Given a graph G(V , E), G 2 is the graph having the same vertex set V and an edge set E : {u, v} ∈ E iff D(u, v) 2 in G.  ...  The difference between radiocoloring and distance-2-coloring in planar graphs The distance-2-coloring problem, discussed above, is formally defined as follows: Definition 8.  ...

### On distance dominator packing coloring in graphs

Jasmina Ferme, Dasa Stesl
2021 Filomat
A coloring c is a distance dominator packing coloring of G if it is a packing coloring of G and for each x ? V(G) there exists i ?  ...  In this paper, we provide some lower and upper bounds on the distance dominator packing chromatic number, characterize graphs G with ?d?(G) ? {2,3}, and provide the exact values of ?d?  ...  Note that if |B 3 | = 1, then t ≤ 7, if |B 3 | = 2, then t ≤ 3, and otherwise, t = 0. G (v), is the maximum distance between v and any other vertex of G: G (v) = max u∈V(G) {d(v, u)}.  ...

### On the Babai and Upper Chromatic Numbers of Graphs of Diameter 2

Peter Johnson, Alexis Krumpelman
2021 Tamkang Journal of Mathematics
In this paper we make progress in the theory of the first Babai number and the upper chromatic number in the simple graph setting, with emphasis on graphs of diameter 2.  ...  The Babai numbers and the upper chromatic number are parameters that can be assigned to any metric space. They can, therefore, be assigned to any connected simple graph.  ...  Since z − t − 1 = r − g(t) − 1, and a color set within which the distance 1 is forbidden must be a subset of one of the parts, after coloring vertices with up to z − t − 1 colors so that the distance 1  ...

### 2-Distance Coloring of Sparse Graphs

Marthe Bonamy, Benjamin Lévêque, Alexandre Pinlou
2014 Journal of Graph Theory
A 2-distance coloring of a graph is a coloring of the vertices such that two vertices at distance at most 2 receive distinct colors.  ...  We prove that every graph with maximum degree ∆ at least 4 and maximum average degree less that 7 3 admits a 2-distance (∆ + 1)-coloring. This result is tight.  ...  Every graph G with mad(G) < 16 7 and ∆(G) ≥ 4 admits a 2-distance (∆(G) + 1)-coloring. Fig. 1 . 1 A graph G with mad(G) = 7 3 , ∆(G) = 4 and χ 2 (G) = 6.  ...

### The Packing Coloring of Distance Graphs D(k,t) [article]

Jan Ekstein, Přemysl Holub, Olivier Togni
2013 arXiv   pre-print
We also give some upper and lower bounds for χ_ρ(D(k, t)) with small k and t. Keywords: distance graph; packing coloring; packing chromatic number  ...  For k < t we study the packing chromatic number of infinite distance graphs D(k, t), i.e. graphs with the set of integers as vertex set and in which two distinct vertices i, j ∈ are adjacent if and only  ...  Remarks and acknowledgemets The access to the METACentrum computing facilities, provided under the programme "Projects of Large Infrastructure for Research, Development and Innovations" LM2010005 funded  ...

### Packing chromatic number of distance graphs

Jan Ekstein, Přemysl Holub, Bernard Lidický
2012 Discrete Applied Mathematics
We study the packing chromatic number of infinite distance graphs G(Z, D), i.e. graphs with the set Z of integers as vertex set and in which two distinct vertices i, j ∈ Z are adjacent if and only if |  ...  In this paper we focus on distance graphs with D = {1, t}. We improve some results of Togni who initiated the study.  ...  Let D(1, t) be a distance graph and B i its i-band.  ...

### 2-distance coloring of sparse graphs

Marthe Bonamy, Benjamin Lévêque, Alexandre Pinlou
2011 Electronic Notes in Discrete Mathematics
A 2-distance coloring of a graph is a coloring of the vertices such that two vertices at distance at most 2 receive distinct colors.  ...  We prove that every graph with maximum degree ∆ at least 4 and maximum average degree less that 7 3 admits a 2-distance (∆ + 1)-coloring. This result is tight.  ...  Every graph G with mad(G) < 16 7 and ∆(G) ≥ 4 admits a 2-distance (∆(G) + 1)-coloring. Fig. 1 . 1 A graph G with mad(G) = 7 3 , ∆(G) = 4 and χ 2 (G) = 6.  ...

### On a metric property of perfect colorings [article]

Anna A. Taranenko
2021 arXiv   pre-print
With the help of an algebraic approach, we deduce corollaries of this result for perfect 2-colorings, perfect colorings in distance-l graphs and in distance-regular graphs.  ...  Given a perfect coloring of a graph, we prove that the L_1 distance between two rows of the adjacency matrix of the graph is not less than the L_1 distance between the corresponding rows of the parameter  ...  One of the most famous examples of such sets and graphs are balls and spheres in distance-regular graphs.  ...
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