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Until recently there were no results on the orientable or nonorientable genus of graphs K n + K m for n / 2 m < n - 1 . Recently, using index one current graphs, McCourt (2014) constructed a nonorientable triangular embedding of K 36 t + 3 + K 18 t + 1 +...

We construct a family of recursive constructions such that for any i { 0 , 1 , 3 , 4 , 6 , 7 , 9 , 10 } and j { 0 , 1 , , 11 } , several arbitrary nonorientable triangular embeddings of every complete graph K m , m i ( mod 12 ) , can be incorporated ...

We describe a class of Skolem sequences of order n such that for the cyclic Steiner triple system of order 6 n + 1 generated by any Skolem sequence from the class we can construct a bipartite index one current graph generating a nonorientable face 2-...

The 1-chromatic number @g"1(S) of a surface S is the maximum chromatic number of all graphs which can be drawn on S so that each edge is crossed by no more than one other edge. It is proved that:(a)There is an integer Q>0 such thatM(N"q)-1=<@g"1(N"q)= =...

We prove that for every n,m>=6, the complete bipartite graph K"n","m has at least 18nm2^@?^(^n^-^1^)^/^3^@?^@?^(^m^-^2^)^/^4^@? nonisomorphic orientable as well as nonorientable genus embeddings.

The vertex-face chromatic number of a map on a surface is the minimum integer m such that the vertices and faces of the map can be colored by m colors in such a way that adjacent or incident elements receive distinct colors. The vertex-face chromatic ...

We give a characterization of a current assignment on the bipartite Mobius ladder graph with 2n+1 rungs. Such an assignment yields an index one current graph with current group Z"1"2"n"+"7 that generates an orientable face 2-colorable triangular ...

We show that for s>=11 there are at least 2^2^s^-^1^1 nonisomorphic orientable triangular embeddings of K"1"2"s"+"3. The result completes the proof that there are constants M, c>0, b>=1/12 such that for every n>=M there are at least c2^b^n nonisomorphic ...

The looseness of a triangular embedding of a complete graph in a closed surface is the minimum integer m such that for every assignment of m colors to the vertices of the embedding (such that all m colors are used) there is a face incident with vertices ...

We prove a theorem that for an integer s>=0, if 12s+7 is a prime number, then the number of nonisomorphic face 3-colorable nonorientable triangular embeddings of K"n, where n=(12s+7)(6s+7), is at least 2^n^^^3^^^/^^^2^(^2^/^7^2^+^o^(^1^)^). By some ...

It was proved earlier that there are constants M, c>0 such that for every n>=M (resp., every n>=M, n@?0,3mod12) there are at least c2^n^/^6 nonisomorphic nonorientable (resp., orientable) genus embeddings of K"n. In the present paper we show that for s>=...

We enumerate all possible trades which involve up to six faces of the face set of a triangular embedding of a simple connected graph. These are classified by the underlying combinatorial trade on the associated block design, and by the geometrical ...