Elementary polycycle

In the case of (5, 3)-polycycles, for every given number we have an example of a (5, 3)-boundary sequence, which admits exactly that number of fillings. The statement and the proof of this theorem used the elementary polycycles presented in Chapter 7 (especially, E, Ci, and C3 from Figure 7.2). 

The decomposition Theorem 7.1.1 (of (r, [Pg.80]

In Figures 7.2 and 7.3, each elementary polycycle is denoted by a certain letter with a subscript two numbers indicate the values of the parameters p, (the number of interior faces) and vint (the number of interior vertices). The infinite series Es, respectively es have (pr, vinl) = (s+2, s), respectively (3s+2, s) and are represented for s < 5, respectively s < 6. 

For every vertex v of an elementary polycycle with n interior vertices, consider all possibilities of adding 2- and 3-gons incident to v, such that the obtained polycycle is elementary and v has become an interior vertex. 

Clearly, if we extend the polycycle along only one of those edges, then the result is not an elementary polycycle. The consideration of all possibilities yields fi and y. Suppose now that P has no infinite components. Then P has at least one infinite... 

We present here applications of elementary polycycle decomposition (in particular, of lists in Figures 7.2 and 7.3) to three problems ... 

Proof. Assume (i) and (ii) hold, and take a closed cycle c in P. The set of elementary polycycles, passed by c, is a finite connected subgraph Majc(P) of Maj(P), so also a tree. If c pass though only one elementary component, then, by (i),... 

Proof The case (r, q) = (3,3), (3,4), (4,3) follows immediately from the list of these polycycles given in Chapter 4. It is clear that doubly infinite and non-periodic (at least in one direction) sequences of glued copies of the elementary polycycles bi and ee, (from Figure 7.3) yield a continuum of infinite non-extensible (3,5> polycycles. By gluing the elementary (5,3)-polycycles Ci (from Figure 7.2) and C 2 (obtained from Ci by rotation through n), we obtain infinite non-extensible (5,3)-polycycles. Clearly, there is a continuum of such. In Lemma 8.2.4, we will construct a continuum of non-extensible (r, )-polycycles for non-elliptic (r, q). 

Consider now the case (r, q) = (5,3). It is easy to see that the only finite elementary (5,3)-polycycle, which is non-extensible, is A = 5, 3 — /. Assume now that Maj(P) has a vertex of degree 1. Then, the elementary polycycle corresponding to this vertex is different from A,. It is easy to see that for all other finite elementary (5,3)-polycycles, we can extend, i.e. add one more face. 

We will shortly present 2-embedding of polycycles just as an application of elementary polycycles. 2-embedding of graphs is the main subject of the book [DGS04]. Let us only mention recent enumeration of 2-embeddable ( a, 6), 3)-spheres (a = 3,4,5). There are 1, 5, 5, respectively, such graphs for a = 3,4,5 (see PDS05, MaSh07]). 

The major skeleton, elementary polycycles, and classification results... 

A (5, 3)-polycycle is called O-elementary if it is elementary and if its boundary sequence is of the form 22391... 223 . By inspection of the list of finite elementaiy (5, 3)-poly cycles (see Figure 7.2), we obtain that there are exactly three O-elementary polycycles Ei, Cu and C3. 

The classification of all Frank-Kasper ( 5, b, 3)-spheres can be done using the elementary polycycles decomposition exposed in Chapter 7. If G is a Frank-Kasper ( 5, b), 3)-sphere, then we remove all its Agonal feces and obtain a (5, 3)gen-polycycle. This (5, 3)ge -polycycle is decomposed into elementary (5,3)gen-polycycles along bridges (see Chapter 7 for definition of those notions). In this chapter a bridge is an edge, where the two vertices are contained in different Agonal feces. 

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