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The edge-face coloring of graphs embedded in a surface of characteristic zero

2009
*
Discrete Mathematics
*

Let G be

doi:10.1016/j.disc.2007.12.055
fatcat:kocmhx3sbfekvdvl2obr3zxyiq
*a**graph**embedded**in**a**surface**of**characteristic**zero*with maximum degree ∆. ... The*edge*-face chromatic number χ e f (G)*of*G is the least number*of**colors*such that any two adjacent*edges*, adjacent faces, incident*edge*and face have different*colors*.*In*this paper, we prove that ... The following is*a*direct result*of*Theorem 17. Corollary 18. If G is*a**graph**embedded**in**a**surface**of**characteristic**zero*with ∆ ≥ 13, then χ e f (G) ≤ ∆ + 1. ...##
###
Coloring edges of graphs embedded in a surface of characteristic zero

2003
*
Journal of combinatorial theory. Series B (Print)
*

Consider

doi:10.1016/s0095-8956(02)00008-4
fatcat:lfbfmfrj4zhulpldbggije7jhe
*graphs*that are embeddable*in**a**surface**of**characteristic**zero*. Class two*graphs**of*this type with maximum degree Dp6 are known to exist. ... These are the first*surfaces*where the maximum degree*of*class two*graphs*has been determined. r ...*in**a**surface**of**characteristic**zero*. ...##
###
On dihedral flows in embedded graphs

2018
*
Journal of Graph Theory
*

Let Γ be

doi:10.1002/jgt.22427
pmid:31217666
pmcid:PMC6559332
fatcat:cb65x6e7yjhtjgkm5funsah5xi
*a*multigraph with for each vertex*a*cyclic order*of*the*edges*incident with it. For n ≥ 3 , let D 2 n be the dihedral group*of*order 2 n . Define D ≔ { ( ± 1*a*0 1 ) ∣*a*∈ Z } . ... Furthermore, the complexity*of*deciding the existence*of*nowhere-identity 2 -flows is discussed. Lastly,*graphs**in*which the equivalence*of*the existence*of*flows as above is true are described. ... We also thank Sven Polak, Lex, and the referees for their valuable comments regarding both the content and the presentation*of*this paper. ...##
###
The Complexity of Bendless Three-Dimensional Orthogonal Graph Drawing

2013
*
Journal of Graph Algorithms and Applications
*

We describe

doi:10.7155/jgaa.00283
fatcat:jabbldkoifegdawpqohuogxjku
*a*correspondence between xyz*graphs*and face-*colored**embeddings**of*the*graph*onto two-dimensional manifolds, and we relate bipartiteness*of*the xyz*graph*to orientability*of*the underlying ... We consider*embeddings**of*3-regular*graphs*into 3-dimensional Cartesian coordinates,*in*such*a*way that two vertices are adjacent if and only if two*of*their three coordinates are equal (that is, if they ... Except as noted, all figures*in*this paper are by the author; all figures remain the copyright*of*their creators and are used by permission. ...##
###
The Topology of Bendless Three-Dimensional Orthogonal Graph Drawing
[chapter]

2009
*
Lecture Notes in Computer Science
*

We define an xyz

doi:10.1007/978-3-642-00219-9_9
fatcat:bwfibbdgdrf3vp7wpiomiv6oz4
*graph*to be*a*spatial*embedding**of**a*3-regular*graph*such that the*edges*at each vertex are mutually perpendicular and no three points lie on an axis-parallel line. ... We describe an equivalence between xyz*graphs*and 3-face-*colored*polyhedral maps, under which bipartiteness*of*the*graph*is equivalent to orientability*of*the map. ... Except as noted, all figures*in*this paper are by the author; all figures remain the copyright*of*their creators and are used by permission. ...##
###
Circular flows of nearly Eulerian graphs and vertex-splitting

2002
*
Journal of Graph Theory
*

0 f S () ¼ 1) such that any

doi:10.1002/jgt.10040
fatcat:mv65xiuy4vekfazkfdwfnpfycq
*graph*G*embedded**in*S with the odd-*edge*connectivity at least f S () admits*a*nowhere-*zero*circular (2 þ )-flow. ... The odd*edge*connectivity*of**a**graph*G, denoted by o (G), is the size*of**a*smallest odd*edge*cut*of*the*graph*. Let S be any given*surface*and be*a*positive real number. ... Lemma 3. 1 . 1 Let G be*a*simple*graph**embedded**in**a**surface*with the Euler*characteristic*c S . ...##
###
On simultaneous colorings of embedded graphs

2000
*
Discrete Mathematics
*

*In*this paper, it is shown that if the maximum degree

*of*

*a*

*graph*is large relative to the genus

*of*the

*embedding*than the

*edge*-face chromatic number

*of*the

*graph*is at most + 1 and its vertex-

*edge*-face ... Discharging This section will use discharging to prove that every

*graph*

*embedded*on

*a*

*surface*

*of*Euler

*characteristic*has one

*of*four special types

*of*elements present

*in*it. ... This article will be concerned with simultaneous

*coloring*problems, that is,

*coloring*some or all

*of*the elements (vertices,

*edges*, and faces)

*of*

*a*

*graph*

*embedded*

*in*

*a*

*surface*so that distinct, adjacent ...

##
###
Variations on Ringel's earth–moon problem

2000
*
Discrete Mathematics
*

*In*particular, we ÿnd

*a*decomposition

*of*K 6n+1 into n toroidal

*graphs*, and from this get

*a*decomposition

*of*K6n into n projective-plane

*graphs*. ... We ÿnd

*a*decomposition

*of*K19 into three Klein bottle

*graphs*and for n ¿ 3, we also conjecture that there exist decompositions

*of*K6n+1 into n Klein bottle

*graphs*. ... Acknowledgements We thank the referees for their many helpful comments and suggestions

*in*revising this paper. ...

##
###
Graph Polynomial for Colored Embedded Graphs: A Topological Approach
[article]

2022
*
arXiv
*
pre-print

We study finite

arXiv:2204.13876v1
fatcat:irwbbmughvesjhizptp7ybyhh4
*graphs**embedded**in*oriented*surfaces*by associating*a*polynomial to it. ... We also analyze*a*variant*of*these polynomials for*colored**embedded**graphs*. This is used to describe the change*in*the polynomial under basic*graph*theoretic operations. ... Patra would like to thank CSIR and IISER Kolkata for funding through*a*research fellowship. ...##
###
Khovano Homology and Embedded Graphs
[article]

2009
*
arXiv
*
pre-print

We construct

arXiv:0911.2608v1
fatcat:3trdyirev5anpe6drrup53ki3e
*a*cobordism group for*embedded**graphs**in*two different ways, first by using sequences*of*two basic operations, called "fusion" and "fission", which*in*terms*of*cobordisms correspond to the ... to*a*such*graph*by using some local replacements at each vertex*in*the*graph*. ... Some parts*of*this paper done*in*Max-Planck Institut für Mathematik (MPIM), Bonn, Germany, the author is kindly would like to thank MPIM for their hosting and subsidy during his study there. ...##
###
Equivalence of edge bicolored graphs on surfaces
[article]

2018
*
arXiv
*
pre-print

Consider the collection

arXiv:1802.03506v1
fatcat:k5mduh5cgvdapm3auv26p3kele
*of**edge*bicolorings*of**a**graph*that is cellularly*embedded*on an orientable*surface*. ...*In*this work, we count the number*of*equivalence classes*of*such*colorings*under two relations: reversing*colors*around*a*face and reversing*colors*around*a*vertex. ... Given*a*checkerboard*coloring**of**a*link diagram lying on the sphere or more generally on*a*compact orientable*surface*,*a*checkerboard*graph**of*the diagram has*a*natural*edge**coloring*coming from crossing ...##
###
Equivalence of Edge Bicolored Graphs on Surfaces

2018
*
Electronic Journal of Combinatorics
*

Consider the collection

doi:10.37236/7384
fatcat:7izt2gozgrhvpdw776pfyxf6kq
*of**edge*bicolorings*of**a**graph*that are cellularly*embedded*on an orientable*surface*. ...*In*this work, we count the number*of*equivalence classes*of*such*colorings*under two relations: reversing*colors*around*a*face and reversing*colors*around*a*vertex. ... Given*a*checkerboard*coloring**of**a*link diagram lying on the sphere or more generally on*a*compact orientable*surface*,*a*checkerboard*graph**of*the diagram has*a*natural*edge**coloring*coming from crossing ...##
###
Page 8940 of Mathematical Reviews Vol. , Issue 2003m
[page]

2003
*
Mathematical Reviews
*

(1-CFL; Orlando, FL)

*Coloring**edges**of**graphs**embedded**in**a**surface**of**characteristic**zero*. (English summary) J. Combin. Theory Ser. B 87 (2003), no. 2, 254-263. ... This paper settles the remaining case by showing that if G is*embedded*on*a**surface**of*Euler*characteristic**zero*and*A*(G) = 7, then 7/(G) =7. ...##
###
Surfaces, Tree-Width, Clique-Minors, and Partitions

2000
*
Journal of combinatorial theory. Series B (Print)
*

*In*general, it is shown that

*a*

*graph*which embeds on

*a*

*surface*

*of*Euler

*characteristic*" has

*a*vertex partition into two

*graphs*

*of*tree-width at most 6 ? ... One

*of*which, that

*a*planar

*graph*may be thus

*edge*partitioned into two series-parallel

*graphs*, has nice generalizations for

*graphs*

*embedded*onto an arbitrary

*surface*and

*graphs*with no large clique-minor ... Let G be

*a*

*graph*

*embedded*on

*a*

*surface*

*of*Euler

*characteristic*

*zero*with the fewest vertices which has no

*edge*partition into two

*graphs*

*of*tree-width at most three. ...

##
###
Islands in Graphs on Surfaces

2016
*
SIAM Journal on Discrete Mathematics
*

*In*this paper, we prove several bounds on the size

*of*islands

*in*large

*graphs*embeddable on fixed

*surfaces*. ... An island

*in*

*a*

*graph*is

*a*set X

*of*vertices, such that each element

*of*X has few neighbors outside X. ... Every

*graph*that has an

*embedding*on

*a*

*surface*

*of*Euler

*characteristic*χ can be

*colored*with

*colors*1, 2, 3, 4, 5,

*in*such

*a*way that each

*color*i 4 induces

*a*

*graph*

*in*which each connected components has ...

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