Quadrangular embeddings of complete graphs

Hartsfield and Ringel proved that a complete graph $K_n$ has an orientable quadrangular embedding if $n\equiv 5 \pmod 8$, and has a nonorientable quadrangular embedding if $n \ge 9$ and $n\equiv 1 \pmod 4$. We complete the characterization of complete graphs admitting quadrangular embeddings by showing that $K_n$ has an orientable quadrilateral embedding if $n \equiv 0 \pmod 8$, and has a nonorientable quadrilateral embedding if $n \equiv 0 \pmod 4$. We also determine the order of minimal quadrangulations for some surfaces where the corresponding graphs are not complete.

[1]  Vladimir P. Korzhik,et al.  On the Number of Nonisomorphic Orientable Regular Embeddings of Complete Graphs , 2001, J. Comb. Theory, Ser. B.

[2]  Bojan Mohar,et al.  Minimal ordered triangulations of surfaces , 1986 .

[3]  Mike J. Grannell,et al.  A lower bound for the number of triangular embeddings of some complete graphs and complete regular tripartite graphs , 2008, J. Comb. Theory, Ser. B.

[4]  Mike J. Grannell,et al.  Exponential Families of Non-Isomorphic Triangulations of Complete Graphs , 2000, J. Comb. Theory, Ser. B.

[5]  G. Ringel.  Der vollständige paare Graph auf nichtorientierbaren Flächen. , 1965 .

[6]  Nora Hartsfield,et al.  Minimal quadrangulations of orientable surfaces , 1989, J. Comb. Theory, Ser. B.

[7]  Arthur T. White,et al.  ON THE GENUS OF THE COMPOSITION OF TWO GRAPHS , 1972 .

[8]  Mike J. Grannell,et al.  Doubly Even Orientable Closed 2-cell Embeddings of the Complete Graph , 2014, Electron. J. Comb..

[9]  David L. Craft,et al.  Surgical techniques for construction minimal orientable imbeddings and joins and compositions of graphs , 1991 .

[10]  Serge Lawrencenko.  Realizing the chromatic numbers and orders of spinal quadrangulations of surfaces , 2012 .

[11]  Ken-ichi Kawarabayashi,et al.  Orientable and Nonorientable Genera for Some Complete Tripartite Graphs , 2004, SIAM J. Discret. Math..

[12]  David L. Craft.  On the genus of joins and compositions of graphs , 1998, Discret. Math..

[13]  Nora Hartsfield,et al.  Minimal quadrangulations of nonorientable surfaces , 1989, J. Comb. Theory, Ser. A.

[14]  G. Ringel.  Das Geschlecht des vollständigen paaren Graphen , 1965 .

[15]  Jonathan L. Gross,et al.  Topological Graph Theory , 1987, Handbook of Graph Theory.

[16]  André Bouchet.  Orientable and nonorientable genus of the complete bipartite graph , 1978, J. Comb. Theory, Ser. B.

[17]  Vladimir P. Korzhik.  Generating Nonisomorphic Quadrangular Embeddings of a Complete Graph , 2013, J. Graph Theory.

[18]  G. Ringel.  Map Color Theorem , 1974 .

[19]  Mike J. Grannell,et al.  A lower bound for the number of orientable triangular embeddings of some complete graphs , 2010, J. Comb. Theory, Ser. B.