Observations on maps and δ-matroids

R. Bruce Richter

Discussiones Mathematicae Graph Theory (1996)

  • Volume: 16, Issue: 2, page 197-205
  • ISSN: 2083-5892

Abstract

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Using a Δ-matroid associated with a map, Anderson et al (J. Combin. Theory (B) 66 (1996) 232-246) showed that one can decide in polynomial time if a medial graph (a 4-regular, 2-face colourable embedded graph) in the sphere, projective plane or torus has two Euler tours that each never cross themselves and never use the same transition at any vertex. With some simple observations, we extend this to the Klein bottle and the sphere with 3 crosscaps and show that the argument does not work in any other surface. We also show there are other Δ-matroids that one can associate with an embedded graph.

How to cite

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R. Bruce Richter. "Observations on maps and δ-matroids." Discussiones Mathematicae Graph Theory 16.2 (1996): 197-205. <http://eudml.org/doc/270187>.

@article{R1996,
abstract = {Using a Δ-matroid associated with a map, Anderson et al (J. Combin. Theory (B) 66 (1996) 232-246) showed that one can decide in polynomial time if a medial graph (a 4-regular, 2-face colourable embedded graph) in the sphere, projective plane or torus has two Euler tours that each never cross themselves and never use the same transition at any vertex. With some simple observations, we extend this to the Klein bottle and the sphere with 3 crosscaps and show that the argument does not work in any other surface. We also show there are other Δ-matroids that one can associate with an embedded graph.},
author = {R. Bruce Richter},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {Δ-matroids; graph embeddings; A-trails},
language = {eng},
number = {2},
pages = {197-205},
title = {Observations on maps and δ-matroids},
url = {http://eudml.org/doc/270187},
volume = {16},
year = {1996},
}

TY - JOUR
AU - R. Bruce Richter
TI - Observations on maps and δ-matroids
JO - Discussiones Mathematicae Graph Theory
PY - 1996
VL - 16
IS - 2
SP - 197
EP - 205
AB - Using a Δ-matroid associated with a map, Anderson et al (J. Combin. Theory (B) 66 (1996) 232-246) showed that one can decide in polynomial time if a medial graph (a 4-regular, 2-face colourable embedded graph) in the sphere, projective plane or torus has two Euler tours that each never cross themselves and never use the same transition at any vertex. With some simple observations, we extend this to the Klein bottle and the sphere with 3 crosscaps and show that the argument does not work in any other surface. We also show there are other Δ-matroids that one can associate with an embedded graph.
LA - eng
KW - Δ-matroids; graph embeddings; A-trails
UR - http://eudml.org/doc/270187
ER -

References

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  1. [1] L.D. Andersen, A. Bouchet and W. Jackson, Orthogonal A-trails of 4-regular graphs embedded in surfaces of low genus, J. Combin. Theory (B) 66 (1996) 232-246, doi: 10.1006/jctb.1996.0017. Zbl0855.05047
  2. [2] A. Bouchet, Maps and Δ-matroids, Discrete Math. 78 (1989) 59-71, doi: 10.1016/0012-365X(89)90161-1. Zbl0719.05019
  3. [3] A. Bouchet, Greedy algorithm and symmetric matroids, Math. Prog. 38 (1987) 147-159, doi: 10.1007/BF02604639. Zbl0633.90089
  4. [4] A. Kotzig, Eulerian lines in finite 4-valent graphs and their transformations, in: Theory of Graphs (P. Erdős and G. Katona, eds.) North-Holland, Amsterdam (1968) 219-230. Zbl0159.54201
  5. [5] R.B. Richter, Spanning trees, Euler tours, medial graphs, left-right paths and cycle spaces, Discrete Math. 89 (1991) 261-268, doi: 10.1016/0012-365X(91)90119-M. Zbl0728.05015
  6. [6] E. Tardos, Generalized matroids and supermodular colorings, in Matroid Theory (Szeged 1982), North- Holland, Amsterdam (1985) 359-382. 
  7. [7] T. Zaslavsky, Biased graphs I, J. Combin. Theory (B) 47 (1989) 32-52, doi: 10.1016/0095-8956(89)90063-4. Zbl0714.05057

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