Chapter I Abstract Embeddings
I.3 Embeddings
Let G = (V, E), V = {v1, v2,· · ·, vn}, be a graph. A point in the
3-dimensional space is represented by a real number t as the parame- ter, e.g., (x, y, z) = (t, t2, t3). Write the vertices as
vi = (xi, yi, zi) = (ti, t2i, t3i)
such that ti 6= tj, i 6= j, 1≤ i, j ≤ n, and an edge as
(u, v) = u+λv, 1≤ λ ≤1,
i.e., the straight line segment between u and v. Because for any four vertices vi, vj, vl and vk, det xi −xj xi −xl xi−xk yi −yj yi −yl yi −yk zi −zj zi −zl zi −zk ! = det ti −tj ti −tl ti−tk t2i −t2j t2i −t2l t2i −t2k t3i −t3j t3i −t3l t3i −t3k
I.3 Embedding 17 = (ti −tj)(ti −tl)(ti −tk)(tk −tl)(tk −tj)(tj −tl) 6= 0, i.e., the four
points are not coplanar, any two edges in G has no intersection inner point.
A representation of a graph on a space with vertices as points and edges as curves pairwise no intersection inner point is called an
embedding of the graph in the space. If all edges are straight line segments in an embedding, then it is called a straight line embedding. Thus, any graph has a straight line embedding in the 3-dimensional space. Similarly, A surface embedding of graph G is a continuous injection µG of an embedding of G on the 3-dimensional space to a surface S such that each connected component of S−µG is homotopic to 0. The connected component is called a face of the embedding. In early books, a surface embedding is also called a cellular embedding. Because only a surface embedding is concerned with in what follows, an embedding is always meant a surface embedding if not necessary to specify.
A graph without circuit is called a tree. A spanning tree of a graph is such a subgraph that is a tree with the same order as the graph. Usually, a spanning tree of a graph is in short called a tree on the graph. For a tree on a graph, the numbers of edges on the tree and not on the tree are only dependent on the order of the graph. They are, respectively, called the rank and the corank of the graph. The corank is also called the Betti number, or cyclic number by some authors.
The following procedure can be used for finding an embedding on a surface.
First, given a cyclic order of all semiedges at each vertex of G, called a rotation. Find a tree(spanning, of course) T on G and distin- guish all the edges not on T by letters. Then, replace each edge not of T by two articulate edges with the same letter.
From this procedure, G is transformed into ˜G without changing the rotation at each vertex except for new vertices that are all ar- ticulate. Because ˜G is a tree, according to the rotation, all lettered articulate edges of ˜G form a polygon with β pairs of edges, and hence
a surface in correspondence with a choice of indices on each pair of the same letter. For convenience, ˜G with a choice of indices of pair in the same letter is called a joint tree of G.
Theorem 1.10 A graphG = (V, E) can always embedded into a surface of orientable genus at most ⌊β/2⌋, or of nonorientable genus at most β, where β is the Betti number of G.
Proof It is seen that any joint tree of G is an embedding of G
on the surface determined by its associate polygon. From (1.8) for the orientable case, the surface has its genus at most ⌊2β/4⌋= ⌊β/2⌋. From (1.9) for the nonorientable case, the surface has its genus at most
2β/2 = β.
In Fig.1.9, graph Gand one of its joint tree are shown. Here, the spanning tree T is represented by edges without letter. a, b and c are edges not on T. Because the polygon is
abcacb ∼top c−1b−1cbaa (Relation 2)
∼top aabbcc (Theorem 1.7),
the joint tree is, in fact, an embedding ofG on a nonorientable surface of genus 3. a b c a a b b c c G G˜
Fig.1.9 Graph and its joint tree
Because any graph with given rotation can always immersed in the plane in agreement with the rotation, each edge has two sides. As known, embeddings of a graph on surfaces are distinguished by the rotation of semiedges at each vertex and the choice of indices of the two semiedges on each edge of the graph whenever edges are labelled
I.3 Embedding 19 by letters. Different indices of the two semiedges of an edge stand for from one side of the edge to the other on a face boundary in an embedding.
Theorem 1.11 A tree can only be embedded on the sphere. Any graphG except tree can be embedded on a nonorientable surface. Any graph G can always be embedded on an orientable surface. Let
nO(G) be the number of distinct embeddings on orientable surfaces,
then the number of embeddings on all surfaces is 2β(G)nO(G), nO(G) =
Y
i≥2
((i−1)!)ni, (1.10)
where β(G) is the Betti number and ni is the number of vertices of
degree i in G.
Proof On a surface of genus not 0, only a graph with at least a circuit is possible to have an embedding. Because a tree has no circuit, it can only embedded on the sphere. Because a graph not a tree has at least one circuit, from Theorem 1.10 the second and the third statements are true. Since distinct planar embeddings of a joint tree of G with the indices of each letter different correspond to distinct embeddings of G on orientable surfaces and the number of distinct planar embeddings of joint trees is
nO(G) =
Y
i≥2
((i−1)!)ni.
Further, since the indices of letters on the β(G) edges has 2β(G) of choices for a given orientable embedding and among them only one choice corresponds to an orientable embedding, the fourth statement
is true.
For an embedding µ(G) of G on a surface, let ν(µG), ǫ(µG) and
φ(µG) are, respectively, its vertex number, or order, edge number, or
size and face number, or coorder.
Theorem 1.12 For a surfaceS, all embeddingsµ(G) of a graph
on S and independent of G. Further, Eul(µG) = 2−2p, p ≥0,
when S has orientable genus p; 2−q, q ≥ 1,
when S has nonorientable genus q.
(1.11)
Proof For an embedding µ(G) on S, if it has at least 2 faces, then by connectedness it has 2 faces with a common edge. From the finite recursion principle, by the inverse of Operation 3 an embedding
µ(G1) of G1 on S with only 1 face on S is found. It is easy to check
that Eul(µG) =Eul(µG1). Similarly, by the inverse of Operation 2 an
embedding µ(G0) of G0 on S with only 1 vertex is found. It is also
easy to check that Eul(µG) =Eul(µG′). Further, by Operation 1 and Relations 1–3, it is seen that Eul(µG0) =Eul(Op), p ≥ 0; or Eul(Qq),
q ≥ 1 according as S is an orientable surface in (1.8); or not in (1.9). From the arbitrariness of G, the first statement is proved.
By calculating the order, size and coorder of Op, p ≥ 0; or Qq,
q ≥ 1, (1.11) is soon obtained. So, the second statement is proved. According to this theorem, for an embedding µ(G) of graph G, Eul(µG) is called its Euler characteristic, or of the surface it is on. Further, g(µG) is the genus of the surface µ(G) is on.
If a graph G is with the minimum length of circuits σ, then from Theorem 1.12 the genus γ(G) of an orientable surface G can be embedded on satisfies the inequality
1− ν(G)−ǫ(1− 2 σ) 2 ≤γ(G) ≤ ⌊ β 2⌋ (1.12)
and the genus ˜γ(G) of a nonorientable surface G can be embedded on satisfies the inequality
2−(ν(G)−ǫ(1− 2
σ)) ≤ γ˜(G) ≤ β. (1.13)
If a graph has an embedding with its genus attaining the lower(upper) bound in (1.12) and (1.13), then it is called down(up)-embeddable. In
I.3 Embedding 21 fact, a graph is up-embeddable on nonorientable, or orientable sur- faces according as it has an embedding with only 1 face, or at most 2 faces.
Theorem 1.13 All graphs but trees are up-embeddable on nonorientable surfaces.
Further, if a graph has an embedding of nonorientable genus l
and an embedding of nonorientable genus k, l < k, then for any i,
l < i < k, it has an embedding of nonorientable genus i.
Proof For an arbitrary embedding of a graph G on a nonori- entable surface, letT be its corresponding joint tree. From the nonori- entability, the associate 2β(G)-gonP has at least 1 letter with different indices(or same power of its two occurrences!). If P = Qq, q = β(G),
then the embedding is an up-embedding in its own right. Otherwise, by Relation 2, or Relation 3 if necessary, whenever s−1s or stst oc- curs, it is, respectively, replaced by ss or sts−1t. In virtue of no letter
missed in the procedure, from the finite recursion principle, P′ = Qq,
q = β(G), is obtained. This is the first statement.
From the arbitrariness of starting embedding in the procedure of proving the first statement by only using Relation 2 instead of Relation 3 (AststB ∼top Ass−1Btt by Relation 2), because the genus
of the surface is increased 1 by 1, the the second statement is true. The second statement of this theorem is also called the inter- polation theorem. The orientable form of interpolation theorem is firstly given by Duke[Duk1]. The maximum(minimum) of the genus of surfaces (orientable or nonorientable) a graph can be embedded on is call the maximum genus(minimum genus) of the graph. The- orem 1.13 shows that graphs but trees are all have their maximum genus on nonorientable surfaces the Betti number with the interpola- tion theorem. The proof would be the simplest one. However, for the orientable case, it is far from simple. many results have been obtained since 1978(see [Liu1–2], [LiuL1], [HuanL1] and [LidL1]) in this aspect. On the determination of minimum genus of a graph, only a few of graphs with certain symmetry are done(see Chapter 12 in [Liu5–6]).