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Improved bounds for some facially constrained colorings
[article]

2020
*
arXiv
*
pre-print

A facial-parity

arXiv:2005.09979v1
fatcat:uz4lpmywjnffvldtjgj5gwspli
*edge*-*coloring**of*a 2-*edge*-connected plane*graph*is a facially-proper*edge*-*coloring**in*which every face is incident*with*zero or an odd number*of**edges**of*each*color*. ... Czap and Jendroľ (*in*Facially-constrained*colorings**of*plane*graphs*: A survey, Discrete Math. 340 (2017), 2691–2703), conjectured that 10*colors*suffice*in**both**colorings*. ...*In*regards*with*facial WORM vertex-*coloring**of*plane*graphs*, it is known that not all plane*graphs*have a (P 3 , P 3 )-WORM*coloring*. ...##
###
Improved bounds for some facially constrained colorings

2020
*
Discussiones Mathematicae Graph Theory
*

A facial-parity

doi:10.7151/dmgt.2357
fatcat:lkmfcv473bfw5ltje6kv5ddct4
*edge*-*coloring**of*a 2-*edge*-connected plane*graph*is a facially-proper*edge*-*coloring**in*which every face is incident*with*zero or an odd number*of**edges**of*each*color*. ... Czap and Jendroľ*in*[Facially-constrained*colorings**of*plane*graphs*: A survey, Discrete Math. 340 (2017) 2691-2703], conjectured that 10*colors*suffice*in**both**colorings*. ... Improved*Bounds*for Some Facially Constrained*Colorings*...##
###
A note on the simultaneous edge coloring
[article]

2020
*
arXiv
*
pre-print

Recently, Cabello raised the following question: given

arXiv:2001.01463v1
fatcat:vhz4nnj4cnbydoy7y7kc2rcduu
*two**graphs*G_1,G_2*of*maximum*degree*Δ on the same set*of*vertices V, is it possible to*edge*-*color*their (*edge*) union*with*Δ+2*colors**in*such a way ... ,G_ℓ*of*maximum*degree*Δ*with*Ω(√(ℓ)·Δ)*colors*and that there exist*graphs*for which this*bound*is tight up to a constant multiplicative factor. ... Since*edges**of*H 2 are*in**both*G 1 and G 2 , the vertex v has*degree*at most ∆ − d*in**both**graphs*H 1 1 and H 2 1 . ...##
###
Parsimonious edge-coloring on surfaces

2018
*
Electronic Journal of Graph Theory and Applications
*

We correct a small error

doi:10.5614/ejgta.2018.6.2.9
fatcat:55bnzvtnsngxxmgrvtowjrteze
*in*a 1996 paper*of*Albertson and Haas, and extend their lower*bound*for the fraction*of*properly*colorable**edges**of*planar subcubic*graphs*that are simple, connected, bridgeless ... , and*edge*-maximal to other surface embeddings*of*subcubic*graphs*. ... Acknowledgement The author is grateful to Mike Albertson for suggesting that she examine this problem, and to*both*Mike Albertson and Ruth Haas for discussions (*both*long-ago and recent)*of*their paper ...##
###
The Game Coloring Number of Planar Graphs

1999
*
Journal of combinatorial theory. Series B (Print)
*

This paper discusses a variation

doi:10.1006/jctb.1998.1878
fatcat:opm6xhs3qrgmthlldvdtqapcme
*of*the game chromatic number*of*a*graph*: the game*coloring*number. This parameter provides an upper*bound*for the game chromatic number*of*a*graph*. ... We show that the game*coloring*number*of*a planar*graph*is at most 19. ... The proof*in*this paper only uses a very simple property*of*planar*graphs*: the existence*of*a``light*edge*''*in*planar*graphs**with*minimum*degree*3, which follows easily from Euler Formula. ...##
###
Improved Bounds for Guarding Plane Graphs with Edges

2019
*
Graphs and Combinatorics
*

*bound*

*of*3n 8

*edges*for any plane

*graph*. 2. ... An

*edge*guard set

*of*a plane

*graph*G is a subset Γ

*of*

*edges*

*of*G such that each face

*of*G is incident to an endpoint

*of*an

*edge*

*in*Γ. Such a set is said to guard G. ... We need to

*color*its vertices

*with*

*two*

*colors*, say white and blue, such that every face contains (i) vertices

*of*

*both*

*colors*and (ii) an

*edge*whose endpoints have the same

*color*. ...

##
###
Improved Bounds for Guarding Plane Graphs with Edges
[article]

2018
*
arXiv
*
pre-print

that G can be guarded

arXiv:1804.07150v1
fatcat:yrda5xoe2zd7dpadcgw3ohqaku
*with*at most 2n/5*edges*, then extend this approach*with*a deeper analysis to yield an improved*bound**of*3n/8*edges*for any plane*graph*. 2- We prove that there exists an*edge*guard ... set*of*G*with*at most n/3+α/9*edges*, where α is the number*of*quadrilateral faces*in*G. ... We need to*color*its vertices*with**two**colors*, say white and blue, such that every face contains (i) vertices*of**both**colors*and (ii) an*edge*whose endpoints have the same*color*. ...##
###
New bounds for locally irregular chromatic index of bipartite and subcubic graphs

2018
*
Journal of combinatorial optimization
*

A

doi:10.1007/s10878-018-0313-7
fatcat:vh3zc5npzbe3dogktfsdapbmo4
*graph*is locally irregular if the neighbors*of*every vertex v have*degrees*distinct from the*degree**of*v. locally irregular*edge*-*coloring**of*a*graph*G is an (improper)*edge*-*coloring*such that the*graph*...*In*addition, we also prove that 4*colors*suffice for locally irregular*edge*-*coloring**of*any subcubic*graph*. ... Lužar also acknowledges partial support by the National Scholarship Programme*of*the Slovak Republic. ...##
###
The Bounded Edge Coloring Problem and Offline Crossbar Scheduling
[article]

2015
*
arXiv
*
pre-print

This paper introduces a variant

arXiv:1512.09002v1
fatcat:omkxexamqfhbhgk7yxw7pmaxlq
*of*the classical*edge**coloring*problem*in**graphs*that can be applied to an offline scheduling problem for crossbar switches. ... We show that the problem is NP-complete, develop three lower*bounds**bounds*on the optimal solution value and evaluate the performance*of*several approximation algorithms,*both*analytically and experimentally ... For each*edge**in*E that is*colored*k − 1, we include a chain*of*three*edges*,*with*the inner*edge*assigned a lower*bound**of*k, while the outer*two**edges*are assigned lower*bounds**of*k − 1. ...##
###
Some results on the palette index of graphs

2019
*
Discrete Mathematics & Theoretical Computer Science
*

Given a proper

doi:10.23638/dmtcs-21-3-11
fatcat:cizc65lcore5tc3lczny75cjdm
*edge**coloring*$\varphi$*of*a*graph*$G$, we define the palette $S_{G}(v,\varphi)$*of*a vertex $v \*in*V(G)$ as the set*of*all*colors*appearing on*edges*incident*with*$v$. ...*In*this paper we give various upper and lower*bounds*on the palette index*of*$G$*in*terms*of*the vertex*degrees**of*$G$, particularly for the case when $G$ is a bipartite*graph**with*small vertex*degrees*... Acknowledgements The authors would like to thank the referees for helpful comments and suggestions, particularly for pointing out an argument which simplified the proof*of*Theorem 3.6. ...##
###
Some results on the palette index of graphs
[article]

2019
*
arXiv
*
pre-print

Given a proper

arXiv:1805.00260v4
fatcat:prrti4o63vc5joqhahqrkttdgm
*edge**coloring*φ*of*a*graph*G, we define the palette S_G(v,φ)*of*a vertex v ∈ V(G) as the set*of*all*colors*appearing on*edges*incident*with*v. ...*In*this paper we give various upper and lower*bounds*on the palette index*of*G*in*terms*of*the vertex*degrees**of*G, particularly for the case when G is a bipartite*graph**with*small vertex*degrees*. ... Acknowledgements The authors would like to thank the referees for helpful comments and suggestions, particularly for pointing out an argument which simplified the proof*of*Theorem 3.6. ...##
###
Properly colored and rainbow copies of graphs with few cherries

2017
*
Journal of combinatorial theory. Series B (Print)
*

We also prove that one can find a rainbow copy

doi:10.1016/j.jctb.2016.07.001
fatcat:2f5makrhizazbir67atnk5wbsa
*of*such G*in*every*edge*-*coloring**of*K_n*in*which all*colors*appear*bounded*number*of*times. ... Let G be an n-vertex*graph*that contains linearly many cherries (i.e., paths on 3 vertices), and let c be a*coloring**of*the*edges**of*the complete*graph*K_n such that at each vertex every*color*appears ... If the*two*other vertices v 1 and v 2*in*P z are*both*neighbors*of*z*in*T 3n , then the vertices*of*P z span a monochromatic cherry and the copy is not properly*edge*-*colored*. ...##
###
The chromatic number of 2-edge-colored and signed graphs of bounded maximum degree
[article]

2020
*
arXiv
*
pre-print

A 2-

arXiv:2009.05439v1
fatcat:v7orrs2u3re25momwmj4qrrbqi
*edge*-*colored**graph*or a signed*graph*is a simple*graph**with**two*types*of**edges*. ... A homomorphism from a 2-*edge*-*colored**graph*G to a 2-*edge*-*colored**graph*H is a mapping φ: V(G) → V(H) that maps every*edge**in*G to an*edge**of*the same type*in*H. ...*in**both*kinds*of**graphs*:*with*a positive or a negative*edge**in*the case*of*2-*edge*-*colored**graphs*,*with*a toward or a backward arc*in*the oriented case. ...##
###
Vertex coloring the square of outerplanar graphs of low degree

2010
*
Discussiones Mathematicae Graph Theory
*

*In*this article we prove that the chromatic number

*of*the square

*of*an outerplanar

*graph*

*of*maximum

*degree*∆ = 6 is 7. ... The optimal upper

*bound*for the chromatic number

*of*the square

*of*an outerplanar

*graph*

*of*maximum

*degree*∆ = 6 is known. ... Acknowledgments The authors are grateful to Steve Hedetniemi for his interest

*in*this problem and for suggesting the writing

*of*this article. ...

##
###
Distributed Algorithms for TDMA Link Scheduling in Sensor Networks

2013
*
International Journal of Networking and Computing
*

*In*this paper, our contribution includes the following: Formulating FDLSP as distance-2

*edge*

*coloring*

*in*bi-directed

*graphs*. ... We formulate the full duplex link scheduling problem (FDLSP) as distance-2

*edge*

*coloring*

*in*bi-directed

*graphs*and prove tighter lower and upper

*bounds*for the FDLSP problem. ... An Upper

*Bound*Lemma 6. Any bi-directed

*graph*G

*with*the maximum out-

*degree*∆ requires at most 2∆ 2

*colors*to distance-2

*edge*-

*color*

*graph*G. Proof. ...

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