A remark on embedded bipartite graphs

Gadi Moran
1987 Journal of combinatorial theory. Series B (Print)  
A use of Euler's formula in Zaks (J. Combin. Theory Ser. B 32 (1982), 95-98 ) is replaced by an elementary argument on permutations. 0 1987 Academic Press, Inc Let 5' be an orientable compact surface, and let G be a finite bipartite graph with u vertices, 2-cell embedded in S. Let m denote the number of 4k-gonal faces (k = 1, 2, 3,...) of this embedding (a face is a connected component of the complement of the graph G in S; G is 2-cell embedded means that every face is homeomorphic to an open
more » ... sc; and a face is k-gonal it is has k edges in its closure). In [Z] Joseph Zaks uses Euler's formula to show that ': and derives as corollaries: (a) A necessary condition for G to have a 2-factor (hence to be hamiltonian) is YYI = 0 (mod 2). (b) If G has no vertex of even degree (in particular, if G is cubic, i.e., 3-valent) then m = 0 (mod 2). We provide here a proof of (*) in which the use of Euler's formula is replaced by parity considerations of three permutations, naturally associated with the embedding. (a) and (b) readily follow from (*) as in [l]. * Supported in part by NSERC grant. ' In [l] (a) is proved only for planar graphs, i.e., when S is the sphere; but in fact Euler's formula yields (*) when S is an arbitrary oriented surface; moreover, it yields also v E m + g (mod 2) if S is a nonorientable compact surface of genus g.
doi:10.1016/0095-8956(87)90023-2 fatcat:pnuqdaflvzfsrokji2jbr5wauu