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A Bijection for Eulerian-equivalence Classes of Totally Cyclic Orientations
2008
Graphs and Combinatorics
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Precisely, the main result of the paper is an algorithmic bijection between the set of Eulerian-equivalence classes of totally cyclic orientations and the set of spanning trees without internally active ...
Gioan showed that the number of cycle reversing classes of totally cyclic orientations of a given graph can be calculated as an evaluation of the corresponding Tutte polynomial. ...
For the bijective proof, we will define a bijection between spanning trees without internally active edges and Eulerian-equivalence classes of totally cyclic orientations. ...
doi:10.1007/s00373-008-0813-9
fatcat:l53ijaw6sba27imesypygvddke
A Blass-Sagan bijection on Eulerian equivalence classes
[article]
2007
arXiv
pre-print
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Following the treatment of Blass and Sagan, we present an algorithmic bijection between the Eulerian equivalence classes of totally cyclic orientations and the spanning trees without internal activity ...
edges for a given graph. ...
The second author would like to thank Professor Beifang Chen for his hospitality during the visit to HKUST. ...
arXiv:0706.3263v1
fatcat:3xkxpsbiyfhi3llch36fmw2poe
A note on flow polynomials of graphs
2009
Discrete Mathematics
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Using the decomposition theory of modular and integral flow polynomials, we answer a problem of Beck and Zaslavsky, by providing a general situation in which the integral flow polynomial is a multiple ...
of the modular flow polynomial. ...
We would like to thank Professor Thomas Zaslavsky for his interest in this work and we are also grateful to two anonymous referees for their valuable comments. ...
doi:10.1016/j.disc.2008.01.050
fatcat:vizrl32mpbdrdmg6xxjhtpgkiu
Orientations, Lattice Polytopes, and Group Arrangements II: Modular and Integral Flow Polynomials of Graphs
2011
Graphs and Combinatorics
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We introduce an Eulerian equivalence relation on orientations, flow arrangements, and flow polytopes; and we apply the theory of Ehrhart polynomials to obtain properties of modular and integral flow polynomials ...
Such viewpoint leads to a reciprocity law for the modular flow polynomial, which gives rise to an interpretation on the values of the modular flow polynomial at negative integers, and answers a question ...
We thank the referees, in particular the second referee, for carefully reading the manuscript and offering several valuable comments. ...
doi:10.1007/s00373-011-1080-8
fatcat:5emfjssliffknlytj3iuls3sqi
Orientations, lattice polytopes, and group arrangements II: Modular and integral flow polynomials of graphs
[article]
2011
arXiv
pre-print
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We introduce an Eulerian equivalence relation on orientations, flow arrangements, and flow polytopes; and we apply the theory of Ehrhart polynomials to obtain properties of modular and integral flow polynomials ...
Such viewpoint leads to a reciprocity law for the modular flow polynomial, which gives rise to an interpretation on the values of the modular flow polynomial at negative integers, and answers a question ...
We thank the referees, in particular the second referee, for carefully reading the manuscript and offering several valuable comments. ...
arXiv:1105.2677v1
fatcat:5gtua5xl2ndsfnpgt5jdt4bgfi
Dual complementary polynomials of graphs and combinatorial interpretation on the values of the Tutte polynomial at positive integers
[article]
2013
arXiv
pre-print
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In particular, some special values of κ_z and κ̅_z (κ and κ̅) count the number of certain special kinds (of equivalence classes) of orientations. ...
We study these polynomials by further introducing a cut-Eulerian equivalence relation on orientations and geometric structures such as the complementary open lattice polyhedron Δ_ctf(G,ϵ), the complementary ...
There are 8 (= T (1, 1)) cut-Eulerian equivalence classes of orientations, 2 (= T (1, 0)) cut equivalence classes of cyclic orientations, and 4 (= T (0, 1)) Eulerian equivalence classes of totally cyclic ...
arXiv:1105.2675v2
fatcat:xi53ed5pbfadloqee2mjzrq7c4
Characterizations of Eulerian and even-face partial duals of ribbon graphs
[article]
2017
arXiv
pre-print
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Then we characterize all Eulerian partial duals of any ribbon graph in terms of crossing-total directions of its medial graph, which are much more simple than semi-crossing directions. ...
Then Metsidik and Jin characterized all Eulerian partial duals of a plane graph in terms of semi-crossing directions of its medial graph. Plane graphs are ribbon graphs with genus 0. ...
Acknowledgements This work is supported by NSFC (No. 11671336) and President's Funds of Xiamen University (No. 20720160011). ...
arXiv:1706.03831v1
fatcat:bqy7ppszivchxisfvxauv2scmu
Generic method for bijections between blossoming trees and planar maps
[article]
2015
arXiv
pre-print
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: for example Eulerian maps, m-Eulerian maps, non separable maps and simple triangulations and quadrangulations of a k-gon. ...
Moreover, it also permits to obtain new bijective constructions for bipolar orientations and d-angulations of girth d of a k-gon. ...
We would like to thankÉric Fusy and Gilles Schaeffer for fruitful discussions and a referee whose careful reading helped us to improve significantly the exposition of our work. ...
arXiv:1305.1312v3
fatcat:axuwgx4esja7flbqll7xvqma5q
Bipartite partial duals and circuits in medial graphs
2013
Combinatorica
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We then characterize all bipartite partial duals of a plane graph in terms of oriented circuits in its medial graph. ...
It is well known that a plane graph is Eulerian if and only if its geometric dual is bipartite. We extend this result to partial duals of plane graphs. ...
Again, at times we abuse notation and identify a ribbon graph with its equivalence class under equality. ...
doi:10.1007/s00493-013-2850-0
fatcat:zn6vdo2r2rdp5gjf7gvhunbqwe
A Generic Method for Bijections between Blossoming Trees and Planar Maps
2015
Electronic Journal of Combinatorics
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: for example Eulerian maps, $m$-Eulerian maps, non-separable maps and simple triangulations and quadrangulations of a $k$-gon. ...
Moreover, it also permits to obtain new bijective constructions for bipolar orientations and $d$-angulations of girth $d$ of a $k$-gon.As for applications, each specialization of the construction translates ...
Acknowledgments We would like to thankÉric Fusy and Gilles Schaeffer for fruitful discussions and a referee whose careful reading helped us to improve significantly the exposition of our work. ...
doi:10.37236/3386
fatcat:vcjsswsberbstfzuvgxttetgxu
EULERIAN AND BIPARTITE ORIENTABLE MATROIDS
[chapter]
2007
Combinatorics, Complexity, and Chance
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The second author extends thanks to NSERC Canada for partial support of this research. We wish to thank D. Archdeacon for contributing a class of Eulerian examples included in this chapter. ...
Acknowledgements The first author would like to thank the Mexican Consejo Nacional de Ciencia y Tecnología (CONACYT) for partially funding this research. ...
Questions remain regarding uniform matroids of rank 3. For example, can one classify Eulerian orientations of U 3,n up to equivalence under e-moves? ...
doi:10.1093/acprof:oso/9780198571278.003.0002
fatcat:3w57alzgirg4bicjvtfpndoc7m
Blossoming bijection for higher-genus maps
[article]
2018
arXiv
pre-print
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In 1997, Schaeffer described a bijection between Eulerian planar maps and some trees. ...
An important step of this construction is to exhibit a canonical orientation for maps, that allows to apply the same local opening algorithm as Schaeffer. ...
Opening non-bicolorable maps We proved in Theorem 3.14 that the opening algorithm, applied to bicolorable maps with dual-geodesic orientation, is actually a bijection with a certain family of unicellular ...
arXiv:1711.05606v3
fatcat:loru5hj3m5gglfla5kn4sod4jq
Bijections and symmetries for the factorizations of the long cycle
[article]
2011
arXiv
pre-print
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These counting results are indicative of a rich combinatorial theory which has remained elusive to this point, and it is the goal of this article to establish a series of bijections which unveil some of ...
Using representation theory, Jackson obtained for each k an elegant formula for counting these factorizations according to the number of cycles of each factor. ...
Consequently, there is a bijection between the set of v 0 -Eulerian tours of G and the set of pairs (A, τ ), where A is a v 0 -arborescence, and τ is an assignment for each vertex v of a total order of ...
arXiv:1112.4970v2
fatcat:g7yz6ugsyvdmznl642x6nc2iay
Bijections and symmetries for the factorizations of the long cycle
2013
Advances in Applied Mathematics
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These counting results are indicative of a rich combinatorial theory which has remained elusive to this point, and it is the goal of this article to establish a series of bijections which unveil some of ...
Using representation theory, Jackson obtained for each k an elegant formula for counting these factorizations according to the number of cycles of each factor. ...
Consequently, there is a bijection between the set of v 0 -Eulerian tours of G and the set of pairs (A, τ ), where A is a v 0 -arborescence, and τ is an assignment for each vertex v of a total order of ...
doi:10.1016/j.aam.2013.01.004
fatcat:vntmvejokfdqxmm3z45dflgjaa
Planar Maps as Labeled Mobiles
2004
Electronic Journal of Combinatorics
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We extend Schaeffer's bijection between rooted quadrangulations and well-labeled trees to the general case of Eulerian planar maps with prescribed face valences to obtain a bijection with a new class of ...
Our bijection covers all the classes of maps previously enumerated by either the two-matrix model used by physicists or by the bijection with blossom trees used by combinatorists. ...
Acknowledgments: All the authors acknowledge the support of the EU network on "Discrete Random Geometry", grant HPRN-CT-1999-00161. ...
doi:10.37236/1822
fatcat:5qllntsep5dv5geurhqrc5ijv4
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