Geodesic
A NOTE ON THE HAMILTONIAN GENUS OF A COMPLETE BIPARTITE GRAPH
Acta mathematica Universitatis Comenianae, Tome 64 (1995) no. 1 
A. Demovic. A NOTE ON THE HAMILTONIAN GENUS OF A COMPLETE BIPARTITE GRAPH. Acta mathematica Universitatis Comenianae, Tome 64 (1995) no. 1. http://geodesic.mathdoc.fr/item/AMUC_1995_64_1_a5/
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     author = {A. Demovic},
     title = {A {NOTE} {ON} {THE} {HAMILTONIAN} {GENUS} {OF} {A} {COMPLETE} {BIPARTITE} {GRAPH}},
     journal = {Acta mathematica Universitatis Comenianae},
     year = {1995},
     volume = {64},
     number = {1},
     url = {http://geodesic.mathdoc.fr/item/AMUC_1995_64_1_a5/}
}
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Voir la notice de l'article provenant de la source Comenius University

The Hamiltonian genus of a graph $G$ (denoted by $\gamma_H(G)$) is the smallest number $g$ such that the graph $G$ is embeddable in the orientable surface with genus $g$ and there is some face-boundary $b$ which is a Hamiltonian cycle of $G$. In this paper we show that \lceil (n-2)(n-1)/4 \rceil \leq \gamma_H(K_n,n) \leq \lceil n/2 \rceil^2 + \lceil n/2 \rceil.