A Kempe switch of a 3-edge-coloring of a cubic graph G on a bicolored cycle C swaps the colors on C and gives rise to a new 3-edge-coloring of G. Two 3-edge-colorings of G are Kempe equivalent if they can be obtained from each other by a sequence of Kempe switches. Fisk proved that any two 3-edge-colorings in a cubic bipartite planar graph are Kempe equivalent. In this paper, we obtain an analog of this theorem and prove that all 3-edge-colorings of a cubic bipartite projective-planar graph G are pairwise Kempe equivalent if and only if G has an embedding in the projective plane such that the chromatic number of the dual triangulation G* is at least 5. As a by-product of the results in this paper, we prove that the list-edge-coloring conjecture holds for cubic graphs G embedded on the projective plane provided that the dual G* is not 4-vertex-colorable.
This is a preview of subscription content, access via your institution.
Buy single article
Instant access to the full article PDF.
Price excludes VAT (USA)
Tax calculation will be finalised during checkout.
T. Abe and K. Ozeki: Signatures of edge-colorings on the projective plane, to appear in Yokohama Math. J.
N. Alon: Restricted colorings of graphs, in: “Surveys in Combinatorics”, Proc. 14 th British Combinatorial Conference, London Mathematical Society Lecture Notes Series 187, edited by K. Walker, Cambridge University Press, 1993, 1–33.
N. Alon, and M. Tarsi: Colorings and orientations of graphs, Combinatorica 12 (1992), 125–134.
K. Appel, and W. Haken: Every planar map is four colorable, Bull. Amer. Math. Soc. 82 (1976), 449–456.
S.-M. Belcastro, and R. Haas: Counting edge-Kempe-equivalence classes for 3-edge-colored cubic graphs, Discrete Math. 325 (2014), 77–84.
B. Bollobás and A.J. Harris: List colorings of graphs, Graphs Combin. 1 (1985), 115–127.
M.E. Bertschi: Perfectly contractile graphs, J. Combin. Theory Ser. B 50 (1990), 222–230.
M. Ellingham, and L. Goddyn: List edge colourings of some 1-factorable multigraphs, Combinatorica 16 (1996), 343–352.
C. Feghali, M. Johnson, and D. Paulusma: Kempe equivalence of colourings of cubic graphs, European J. Combin. 59 (2017), 1–10.
S. Fisk: Comninatorial structure on triangulations. I. The structure of four colorings, Advances in Math. 11 (1973), 326–338.
S. Fisk: Combinatorial structure on triangulations. II. Local colorings, Advances in Math. 11 (1973), 339–358.
S. Fisk: Geometric coloring theory, Advances in Math. 324 (1977), 298–340.
I.J. Holyer: The NP-completeness of edge colourings, SIAM J. Comput. 10 (1980), 718–720.
F. Jaeger: On the Penrose number of cubic diagrams, Discrete Math. 74 (1989), 85–97.
J. Karabáš, E. Máčajová, AND R. Nedela: 6-decomposition of snarks, European J. Combin. 34 (2013), 111–122.
L.H. Kauffman: Map coloring and the vector cross product, J. Combin. Theory Ser. B 48 (1990), 145–154.
L.H. Kauffman: Reformulating the map color theorem, Discrete Math. 302 (2005), 145–172.
A.B. Kempe: On the Geographical Problem of Four-Colors, Amer. J. Math. 2 (1879), 193–200.
M. Kobayashi, A. Nakamoto, and T. Yamaguchi: Polychromatic 4-coloring of cubic even embeddings on the projective plane, Discrete Math. 313 (2013), 2423–2431.
A. Kündgen, and C. Thomassen: Spanning quadrangulations of triangulated surfaces, Abh. Math. Semin. Univ. Hambg. 87 (2017), 357–368.
R. Lukot’ka, and E. Rollova: Perfect matchings of regular bipartite graphs, J. Graph Theory 85 (2017), 525–532.
E. Máčajová, AND M. Škoviera: Irreducible snarks of given order and cyclic connectivity, Discrete Math. 306 (2006), 779–791.
J. McDonald, B. Mohar, and D. Scheide: Kempe Equivalence of Edge-Colorings in Subcubic and Subquartic Graphs, J. Graph Theory 70 (2012), 226–239.
B. Mohar: Coloring Eulerian triangulations of the projective plane, Discrete Math. 244 (2002), 339–343.
B. Mohar: Kempe equivalence of colorings, Graph Theory Trends in Mathematics, (2006) 287–297.
B. Mohar, and J. Salas: A new Kempe invariant and the (non)-ergodicity of the Wang-Swendsen-Kotecky algorithm, J. Phys. A: Math. Theor. 42 (2009), 225204.
B. Mohar, and C. Thomassen: Graphs on Surfaces, Johns Hopkins Univ. Press, Baltimore, (2001).
A. Nakamoto, K. Noguchi, and K. Ozeki: Spanning bipartite quadrangulations of even triangulations, J. Graph Theory 90 (2019), 267–287.
R. Nedela, and M. Škoviera: Decompositions and reductions of snarks, J. Graph Theory 22 (1996), 253–279.
S. Norine, and R. Thomas: Pfaffian labelings and signs of edge colorings, Combinatorica 28 (2008), 99–111.
Y. Nozaki: Personal communication, 2020.
N. Robertson, D. Sanders, P.D. Seymour, and R. Thomas: The four-color theorem, J. Combin. Theory Ser. B 70 (1997), 2–44.
D.E. Scheim: The number of edge 3-colorings of a planar cubic graph as a permanent, Discrete Math. 8 (1974), 377–382.
E. Steffen: Classifications and characterizations of snarks, Discrete Math. 188 (1998), 183–203.
Y. Suzuki, and T. Watanabe: Generating even triangulations of the projective plane, J. Graph Theory 56 (2007), 333–349.
P.G. Tait: Remarks on the colourings of maps, Proc. R. Soc. Edinburgh 10 (1880), 729.
R. Thomas: A survey of Pfaffian orientations of graphs, International Congress of Mathematicians III (2006), 963–984.
M.L. Vergnas, and H. Meyniel: Kempe classes and the Hadwiger conjecture, J. Combin. Theory Ser. B 31 (1981), 95–104.
E. Vigoda: Improved bounds for sampling colorings, J. Math. Phys. 41 (2000), 1555–1569.
J.-S. Wang, R.H. Swendsen, and R. Kotecký: Antiferromagnetic Potts models, Phys. Rev. Lett. 63 (1989), 109–112.
The author thanks the anonymous reviewers for carefully reading the paper and for their helpful comments, which considerably improved both the content and the readability of the paper. The author is also grateful to Yuta Nozaki, who constructed cubic bipartite graphs embedded on the projective plane having more than three Kempe equivalence classes, as in Figure 13. This work was supported by JSPS KAKENHI, Grant Numbers 18K03391 and 20H05795, and the Research Institute for Mathematical Sciences, an International Joint Usage/Research Center located in Kyoto University.
Dedicated to Professor Katsuhiro Ota on the occasion of his 60th birthday
Rights and permissions
About this article
Cite this article
Ozeki, K. Kempe Equivalence Classes of Cubic Graphs Embedded on the Projective Plane. Combinatorica 42 (Suppl 2), 1451–1480 (2022). https://doi.org/10.1007/s00493-021-4330-2
Mathematics Subject Classification (2010)