PDF

Discussiones Mathematicae Graph Theory 16(2) (1996) 197-205
DOI: https://doi.org/10.7151/dmgt.1034

OBSERVATIONS ON MAPS AND Δ-MATROIDS

R. Bruce Richter

Department of Mathematics and Statistics, Carleton University
Ottawa Canada K1H 8H1
email: [email protected]

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.

Keywords: Δ-matroids, graph embeddings, A-trails.

1991 Mathematics Subject Classification: 05C10, 05B35.

References

[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.
[2]	A. Bouchet, Maps and Δ-matroids, Discrete Math. 78 (1989) 59-71, doi: 10.1016/0012-365X(89)90161-1.
[3]	A. Bouchet, Greedy algorithm and symmetric matroids, Math. Prog. 38 (1987) 147-159, doi: 10.1007/BF02604639.
[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.
[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.
[6]	E. Tardos, Generalized matroids and supermodular colorings, in Matroid Theory (Szeged 1982), North- Holland, Amsterdam (1985) 359-382.
[7]	T. Zaslavsky, Biased graphs I, J. Combin. Theory (B) 47 (1989) 32-52, doi: 10.1016/0095-8956(89)90063-4.