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

Weifan Wang
2009 Discrete Mathematics  
Let G be 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.  ... 
doi:10.1016/j.disc.2007.12.055 fatcat:kocmhx3sbfekvdvl2obr3zxyiq

Coloring edges of graphs embedded in a surface of characteristic zero

Daniel P. Sanders, Yue Zhao
2003 Journal of combinatorial theory. Series B (Print)  
Consider 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.  ... 
doi:10.1016/s0095-8956(02)00008-4 fatcat:lfbfmfrj4zhulpldbggije7jhe

On dihedral flows in embedded graphs

Bart Litjens
2018 Journal of Graph Theory  
Let Γ be 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.  ... 
doi:10.1002/jgt.22427 pmid:31217666 pmcid:PMC6559332 fatcat:cb65x6e7yjhtjgkm5funsah5xi

The Complexity of Bendless Three-Dimensional Orthogonal Graph Drawing

David Eppstein
2013 Journal of Graph Algorithms and Applications  
We describe 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.  ... 
doi:10.7155/jgaa.00283 fatcat:jabbldkoifegdawpqohuogxjku

The Topology of Bendless Three-Dimensional Orthogonal Graph Drawing [chapter]

David Eppstein
2009 Lecture Notes in Computer Science  
We define an xyz 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.  ... 
doi:10.1007/978-3-642-00219-9_9 fatcat:bwfibbdgdrf3vp7wpiomiv6oz4

Circular flows of nearly Eulerian graphs and vertex-splitting

Cun Quan Zhang
2002 Journal of Graph Theory  
0 f S () ¼ 1) such that any 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 .  ... 
doi:10.1002/jgt.10040 fatcat:mv65xiuy4vekfazkfdwfnpfycq

On simultaneous colorings of embedded graphs

Daniel P. Sanders, John Maharry
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  ... 
doi:10.1016/s0012-365x(00)00064-9 fatcat:7z65ori76bdqvbc5fitlwe62dq

Variations on Ringel's earth–moon problem

Brad Jackson, Gerhard Ringel
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.  ... 
doi:10.1016/s0012-365x(99)00278-2 fatcat:j656yihqyrddtankdx3bl65ree

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

Somnath Basu, Dhruv Bhasin, Siddhartha Lal, Siddhartha Patra
2022 arXiv   pre-print
We study finite 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.  ... 
arXiv:2204.13876v1 fatcat:irwbbmughvesjhizptp7ybyhh4

Khovano Homology and Embedded Graphs [article]

Ahmad Zainy Al-Yasry
2009 arXiv   pre-print
We construct 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.  ... 
arXiv:0911.2608v1 fatcat:3trdyirev5anpe6drrup53ki3e

Equivalence of edge bicolored graphs on surfaces [article]

Oliver T. Dasbach, Heather M. Russell
2018 arXiv   pre-print
Consider the collection 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  ... 
arXiv:1802.03506v1 fatcat:k5mduh5cgvdapm3auv26p3kele

Equivalence of Edge Bicolored Graphs on Surfaces

Oliver T. Dasbach, Heather M. Russell
2018 Electronic Journal of Combinatorics  
Consider the collection 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  ... 
doi:10.37236/7384 fatcat:7izt2gozgrhvpdw776pfyxf6kq

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

Guoli Ding, Bogdan Oporowski, Daniel P. Sanders, Dirk Vertigan
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.  ... 
doi:10.1006/jctb.2000.1962 fatcat:j7hw2spwajeaparmwn4uo4voae

Islands in Graphs on Surfaces

Louis Esperet, Pascal Ochem
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  ... 
doi:10.1137/140957883 fatcat:5qnkqk6p7vfu7ocvpenzaprowq
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