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Distance graphs and the T-coloring problem

1999
*
Discrete Mathematics
*

Exploiting an equivalence between the complete

doi:10.1016/s0012-365x(98)00199-x
fatcat:ejws7v3ddvhxxng3cckexyykxi
*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*...##
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Distance-Two Colorings of Barnette Graphs
[article]

2018
*
arXiv
*
pre-print

We fully describe all Goodey

arXiv:1807.01410v1
fatcat:bdizkx526jej5lhmpy6fk7pu7m
*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 . ...##
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Parameterized Coloring Problems on Threshold Graphs
[article]

2020
*
arXiv
*
pre-print

We show that Precoloring Extension is fixed-parameter tractable (FPT) parameterized by

arXiv:1910.10364v3
fatcat:gvi4gqg2o5gvpjk6ovthtgqyzm
*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*. ...##
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Chromatic Coloring of Distance Graphs I

2021
*
VOLUME-8 ISSUE-10, AUGUST 2019, REGULAR ISSUE
*

We prove some interesting results related to the computation of chromatic number of certain

doi:10.35940/ijitee.i9291.0710921
fatcat:mc3qomlgbvevhkra66ksr7ppum
*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 ...##
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Distributed self-stabilizing placement of replicated resources in emerging networks

2005
*
IEEE/ACM Transactions on Networking
*

Our combination of theoretical results

doi:10.1109/tnet.2005.850196
fatcat:p64elbi7ynewlfk4fbatro7p3i
*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 ...##
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Open Distance Pattern Coloring of a Graph

2012
*
Journal of Fuzzy Set Valued Analysis
*

Case 2: Center of

doi:10.5899/2012/jfsva-00144
fatcat:mqurqxkprzfx7d7no6qvhdvify
*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*. ...##
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Solutions of some L(2,1)-coloring related open problems

2016
*
Discussiones Mathematicae Graph Theory
*

An L(2, 1)-

doi:10.7151/dmgt.1858
fatcat:k5az5k7qofbthd4bwt65xoden4
*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 ...##
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Radiocoloring in planar graphs: Complexity and approximations

2005
*
Theoretical Computer Science
*

The FAP is usually modelled by variations of the

doi:10.1016/j.tcs.2005.03.013
fatcat:2ojjviuilzdk7mbmw6er4ksrqu
*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. ...##
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On distance dominator packing coloring in graphs

2021
*
Filomat
*

A

doi:10.2298/fil2112005f
fatcat:4djhqgvfrvap7mtgvfqa627klu
*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)}. ...##
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On the Babai and Upper Chromatic Numbers of Graphs of Diameter 2

2021
*
Tamkang Journal of Mathematics
*

In this paper we make progress in the theory of the first Babai number

doi:10.5556/j.tkjm.52.2021.3430
fatcat:vpsgeivxbrecnen2z3k2lz2kze
*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 ...##
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2-Distance Coloring of Sparse Graphs

2014
*
Journal of Graph Theory
*

A 2-

doi:10.1002/jgt.21782
fatcat:lqkozalajjbzfhrgypsrztrbu4
*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. ...##
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The Packing Coloring of Distance Graphs D(k,t)
[article]

2013
*
arXiv
*
pre-print

We also give some upper

arXiv:1302.0721v1
fatcat:ykr7i5ni5bbevgf6cpct5xemly
*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 ...##
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Packing chromatic number of distance graphs

2012
*
Discrete Applied Mathematics
*

We study the packing chromatic number of infinite

doi:10.1016/j.dam.2011.11.022
fatcat:vfqr72ch2rb5jmzu3sb2e4u2rq
*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. ...##
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2-distance coloring of sparse graphs

2011
*
Electronic Notes in Discrete Mathematics
*

A 2-

doi:10.1016/j.endm.2011.09.027
fatcat:6yerpnfgqrcltoabmwlclmh4ii
*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. ...##
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On a metric property of perfect colorings
[article]

2021
*
arXiv
*
pre-print

With the help of an algebraic approach, we deduce corollaries of this result for perfect 2-

arXiv:2102.01958v1
fatcat:t3lyykal4jeknjzas2i4btrsvm
*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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