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11. The construction of a torus from the unit square.  

11. The construction of a torus from the unit square.  

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A prominent question of topological graph theory is "what type of surface can a nonplanar graph be embedded into?" This thesis has two main goals. First to provide a necessary background in topology and graph theory to understand the development of an embedding algorithm. The main purpose is developing and proving a direct constructive embedding al...

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... For example, let us take the rotation system on Q 3 that we dened previously and reverse the ordering of the edges around vertex 0. We change the rotation system ordering from (1,2,4) to (1,4,2) and the new rotation system is now Let us use the boundary walk algorithm to nd the boundary walks of this embedding. Since the entry around 0 has changed, when we start the walk with 0 − 1, and reach vertex 0 from 4, instead of going towards 1 and closing this boundary walk, we now go towards 2 and continue for a while before this boundary walk closes up. ...
... For example, let us take the rotation system on Q 3 that we dened previously and reverse the ordering of the edges around vertex 0. We change the rotation system ordering from (1,2,4) to (1,4,2) and the new rotation system is now Let us use the boundary walk algorithm to nd the boundary walks of this embedding. Since the entry around 0 has changed, when we start the walk with 0 − 1, and reach vertex 0 from 4, instead of going towards 1 and closing this boundary walk, we now go towards 2 and continue for a while before this boundary walk closes up. ...
... A convex icosahedron GRAPH EMBEDDING INTO SURFACESIn this chapter we introduce the main concepts and results related to graph embeddings and their representations ia rotation systems. The following are references for the material in sections 2.1-2.3 :[4] ...
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The purpose of this thesis is to study hypercube graphs and their embeddings on orientable surfaces. We use rotation systems to represent these embeddings. We prove some results about the effect of adjacent switches in rotation system and create a rotation system called the ABC rotation system and prove general results about it. Using this rotation system, we give a general theorem about the minimal embedding of $Q_{n}$. We also look at some interesting types of maximal embedding of $Q_{n}$, such as the Eulerian walk embedding and the "big-face embedding". We prove a theorem that gives a recursively constructive way of obtaining the latter embedding in $Q_{n}