Abstract
A graph G is said to be equimatchable if every matching in G extends to (i.e., is a subset of) a maximum matching in G. In an earlier paper with Saito, the authors showed that there are only a finite number of 3-connected equimatchable planar graphs. In the present paper, this result is extended by showing that in a surface of any fixed genus (orientable or non-orientable), there are only a finite number of 3-connected equimatchable graphs having a minimal embedding of representativity at least three. (In fact, if the graphs considered are non-bipartite, the representativity three hypothesis may be dropped.) The proof makes use of the Gallai-Edmonds decomposition theorem for matchings.
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Aldred, R.E.L., Kawarabayashi, K., Plummer, M.D.: On the matching extendability of graphs in surfaces. J. Combin. Theory Ser. B 98, 105–115 (2008)
Barnette, D.: “Polyhedral maps on 2-manifolds,” Convexity and Related Combinatorial Geometry (Norman, Oklahoma, 1980), New York (1982), pp. 7–19
Edmonds, J.: Paths, trees and flowers. Canad. J. Math. 17, 449–467 (1965)
Gallai, T.: Kritische Graphen II. Magyar Tud. Akad. Mat. Kutató Int. Közl. 8, 373–395 (1963)
Gallai, T.: Maximale Systeme unabhängiger Kanten. Magyar Tud. Akad. Mat. Kutató Int. Közl. 9, 401–413 (1964)
Kawarabayashi, K., Plummer, M.D., Saito, A.: On two equimatchable graph classes. Discrete Math. 266, 263–274 (2003)
Lebesgue, H.: Quelques conséquences simples de la formule d’Euler. J. Math. 9, 27–43 (1940)
Lesk, M., Pulleyblank, W.R., Plummer, M.D.: “Equimatchable graphs”, Graph Theory and Combinatorics (Proceedings of Cambridge Conference in Honor of Paul Erdős) B. Bollobás (Editor), Academic Press, London (1984), pp. 239–254
Lewin, M.: Matching-perfect and cover-perfect graphs. Israel J. Math. 18, 345–347 (1974)
Lovász, L., Plummer, M.D.: Matching Theory, Akadémiai Kiadó Budapest and Annals of Discrete Mathematics, Vol. 29, North-Holland, Amsterdam, 1986
Meng, D.H.-C.: Matchings and coverings for graphs, PhD thesis, Michigan State University, East Lansing, 1974
Mohar, B.: Existence of polyhedral embeddings of graphs. Combinatorica 21, 395–401 (2001)
Mohar, B., Thomassen, C.: Graphs on Surfaces. Johns Hopkins Univ. Press, Baltimore and London, (2001)
Parsons, T., Pica, G., Pisanski, T., Ventre, A.: Orientably simple graphs. Math. Slovaca 37, 391–394 (1987)
Plummer, M.: On n-extendable graphs. Discrete Math. 31, 201–210 (1980)
Ringel, G.: Das Geschlecht des vollständigen paaren Graphen. Math. Sem. Univ. Hamburg 28, 139–150 (1965)
Ringel, G.: Der vollständige paare Graph auf nichtorientierbaren Flächen. J. Reine Angew. Math. 220, 89–93 (1965)
Ringel, G., Youngs, J.W.T.: Solution of the Heawood map-coloring problem. Proc. Nat. Acad. Sci. U.S.A. 60, 438–445 (1968)
Robertson, N., Vitray, R.: “Representativity of surface embeddings”, Paths, Flows and VLSI-Layout B. Korte et al. (Editors), Springer-Verlag, Berlin (1990), 293–328
Sumner, D.P.: Randomly matchable graphs. J. Graph Theory 3, 183–186 (1979)
Thomassen, C.: Tilings of the torus and the Klein bottle and vertex-transitive graphs on a fixed surface. Trans. Amer. Math. Soc. 323, 605–635 (1991)
Youngs, J.W.T.: Minimal imbeddings and the genus of a graph. J. Math. Mech. 12, 303–315 (1963)
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Kawarabayashi, Ki., Plummer, M.D. Bounding the Size of Equimatchable Graphs of Fixed Genus. Graphs and Combinatorics 25, 91–99 (2009). https://doi.org/10.1007/s00373-008-0838-0
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DOI: https://doi.org/10.1007/s00373-008-0838-0