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A Bijection for Eulerian-equivalence Classes of Totally Cyclic Orientations

Beifang Chen, Arthur L. B. Yang, Terence Y. J. Zhang
2008 Graphs and Combinatorics  
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]

Beifang Chen, Arthur L. B. Yang, Terence Y. J. Zhang
2007 arXiv   pre-print
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

Beifang Chen, Arthur L.B. Yang
2009 Discrete Mathematics  
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

Beifang Chen, Richard P. Stanley
2011 Graphs and Combinatorics  
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]

Beifang Chen
2011 arXiv   pre-print
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]

Beifang Chen
2013 arXiv   pre-print
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]

Qingying Deng, Xian'an Jin
2017 arXiv   pre-print
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]

Marie Albenque, Dominique Poulalhon
2015 arXiv   pre-print
: 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

Stephen Huggett, Iain Moffatt
2013 Combinatorica  
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

Marie Albenque, Dominique Poulalhon
2015 Electronic Journal of Combinatorics  
: 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]

Laura E. Chávez Lomelí, Luis A. Goddyn
2007 Combinatorics, Complexity, and Chance  
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]

Mathias Lepoutre
2018 arXiv   pre-print
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]

Olivier Bernardi
2011 arXiv   pre-print
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

Olivier Bernardi, Alejandro H. Morales
2013 Advances in Applied Mathematics  
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

J. Bouttier, P. Di Francesco, E. Guitter
2004 Electronic Journal of Combinatorics  
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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