We give explicit deformations of embeddings of abstractly planar graphs that lie on the standard torus T 2 ⊂ R3 and that contain neither a nontrivial knot nor a nonsplit link into the plane. It follows that ravels do not embed on the torus. Our results provide general insight into properties of molecules that are synthesized on a torus.

We show that a 6-regular graph of diameter d embedded on the torus can have at most 3d + 3d+ 1 vertices having exhibited a graph of this order for each d ≥ 1. 1 6-regular Toroidal Graphs In [1] it is proven that all 6-regular toroidal graphs can have their vertices arranged in a rectangular based grid structure with parallel diagonals across every sub-rectangle in the grid. A special case of th...

The balanced Hamiltonian cycle problem is a quiet new topic of graph theorem. Given a graph G = (V, E), whose edge set can be partitioned into k dimensions, for positive integer k and a Hamiltonian cycle C on G. The set of all i-dimensional edge of C, which is a subset by E(C), is denoted as Ei(C). If ||Ei(C)| |Ej(C)|| 1 for 1 i < j k, C is called a balanced Hamiltonian cycle. In this paper, th...

An (L, d)∗-coloring is a mapping φ that assigns a color φ(v) ∈ L(v) to each vertex v ∈ V (G) such that at most d neighbors of v receive colore φ(v). A graph is called (m, d)∗-choosable, if G admits an (L, d)∗-coloring for every list assignment L with |L(v)| ≥ m for all v ∈ V (G). In this note, it is proved that every toroidal graph, which contains no adjacent triangles and contains no 6-cycles ...

1 We investigate the toroidal expanse of an embedded graph G, that is, the size of the largest 2 toroidal grid contained in G as a minor. In the course of this work we introduce a new embedding 3 density parameter, the stretch of an embedded graph G, and use it to bound the toroidal 4 expanse from above and from below within a constant factor depending only on the genus and 5 the maximum degree...

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A longest sequence S of distinct vertices of a graph G such that each vertex of S dominates some vertex that is not dominated by its preceding vertices, is called a Grundy dominating sequence; the length of S is the Grundy domination number of G. In this paper we study the Grundy domination number in the four standard graph products: the Cartesian, the lexicographic, the direct, and the strong ...

We determine upper bounds for γ(Qn) and i(Qn), the domination and independent domination numbers, respectively, of the graph Qn obtained from the moves of queens on the n× n chessboard drawn on the torus.

An anti-magic labeling of a finite simple undirected graph with p vertices and q edges is a bijection from the set of edges to the set of integers {1, 2, ..., q} such that the vertex sums are pairwise distinct, where the vertex sum at vertex u is the sum of labels of all edges incident to such vertex. A graph is called anti-magic if it admits an antimagic labeling. Hartsfield and Ringel conject...