ryo nikkuni- regular projections of spatial graphs

8/3/2019 Ryo Nikkuni- Regular Projections of Spatial Graphs

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Regular Projections of Spatial Graphs

Ryo NIKKUNI

ABSTRACT

Study of the regular projection in spatial graph theory is more rich in content as compared withthe regular projection in knot theory. Actually some interesting phenomena in the regular projectionof spatial graphs which are not appeared in the one of knots and links have been discovered Since1990s. In this article we introduce the recent topics in the regular projection of spatial graphs,in particular, knotted projection (KP), identifiable projection (IP) and completely distinguishableprojection (CDP). We also present some unsolved problems in this area.

1. Introduction

1.1. KP, IP and CDP

Let G be a finite graph. We give a label to each of vertices and edges of G and denote

the set of all vertices and the set of all edges of a graph G by V(G) and E(G), respectively.

An embedding of G into R3 is called a spatial embedding of G or simply a spatial graph.

Two spatial embeddings f and g of G are said to be ambient isotopic if there exists an

orientation preserving homeomorphism : R3 R3 such that f = g. A graph G

is said to be planar if there exists an embedding of G into R2, and a spatial embedding

of a planar graph G is said to be trivial if it is ambient isotopic to an embedding of G

into R2 R3. We note that a trivial spatial embedding of a planar graph is unique up

to ambient isotopy in R3 [14].

A regular projectionof G is an immersion from G to R2 whose multiple points are only

finitely many transversal double points away from vertices. For a regular projection f of

G with p double points, we can obtain 2p regular diagrams of the spatial embeddings of

G from f by giving over/under information to each of the double points. Then we say

that a spatial embedding f of G is obtained from f if f is represented by one of these 2p

regular diagrams. At this time we also call f a regular projection of f.

Let f be a regular projection of a graph which is homeomorphic to the disjoint union

of 1-spheres. Then it is well known that a trivial spatial embedding, namely a trivial

link, can be always obtained from f. This fact plays an important role in knot theory,

e.g. the theory of skein polynomial invariants is based on this fact. But this fact does

not always hold for a regular projection of a planar graph. For example, let G be the

octahedron graph and f a regular projection of G as illustrated in Fig. 1.1.1. Taniyama

pointed out that any of the spatial embeddings of G obtained from f is non-trivial [27].

Such a regular projection is said to be knotted, namely a regular projection

f of a planar

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graph G is said to be knotted [27] if any of the spatial embeddings of G obtained from

f is non-trivial. We call a knotted regular projection a KP simply. The existence of a

KP is one of the suggestive phenomena in spatial graph theory. In the next section, we

introduce some fundamental results and the recent development in the study of KP.

f

3

G

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2

1

6

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Fig. 1.1.1. A knotted regular projection

While a KP detects a non-triviality of a regular projection, let us consider a unifor-

mity of a regular projection. A regular projection f of G is said to be identifiable [10] if

all regular diagrams of the spatial embeddings of G obtained from f represent mutually

ambient isotopic spatial embeddings of G. For example, each of the regular projections

as illustrated in Fig. 1.1.2 (1), (2) and (3) is identifiable. Actually we can see that only

trivial spatial embeddings can be obtained from each of these regular projections. We

call an identifiable regular projection an IP simply. As the examples above suggest, an IPdetects a triviality of a regular projection. Namely, an IP cannot be a KP. We explain

the details in section 3.

(1) (2) (3)

1 2

1

4 3

2

Fig. 1.1.2. Identifiable regular projections

On the other hand, a regular projection f of G is said to be completely distinguishable

[17] if all regular diagrams of the spatial embeddings of G obtained from f represent

mutually different spatial embeddings of G up to ambient isotopy. For example, let us

recall a regular projection f of the octahedron graph as illustrated in Fig. 1.1.1. As we

said before,f

is a KP, namely we can obtain the eight spatial embeddings ofG

from

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f as illustrated in Fig. 1.1.3 and it can be checked that they are non-trivial spatial

embeddings. Moreover, by observing the constituent knots and links carefully, we can

see that these eight spatial embeddings are mutually not ambient isotopic. Thus f is

completely distinguishable. We call a completely distinguishable regular projection aCDP simply. As the example above suggests, a CDP of a planar graph is related to

KP. Moreover, a CDP of a non-planar graph can be also considered and relates to the

minimal crossing number of a graph. In section 4, we introduce the recent results in the

study of CDP.

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Fig. 1.1.3.

2. Knotted projection (KP)

2.1. Trivializable graphs

A planar graph is said to be trivializable if the graph does not have a KP. For example, a

graph homeomorphic to the disjoint union of 1-spheres is trivializable, and the octahedron

graph is not trivializable.

Question 2.1.1. When is a planar graph trivializable?

It is known that there exist infinitely many trivializable graphs. Taniyama showed that

every bifocal as illustrated on the left-hand side in Fig. 2.1.1 is trivializable [27]. On

the other hand, Sugiura and Suzuki showed that every 3-line web as illustrated on the

right-hand side in Fig. 2.1.1 is trivializable [23]. Tamura showed that every neo-bifocalis

trivializable and gave a systematic construction of trivializable graphs in terms of an edge

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sum of graphs [24].1 We refer the reader to [27], [24] and [23] for the precise definitions

of the bifocal, the neo-bifocal and the 3-line web, respectively. But trivializable graphs

have not been characterized completely yet.

i

Fig. 2.1.1. A bifocal and a 3-line web

2.2. Forbidden graphs for the trivializability

We investigate trivializable graphs from a stand point of graph minor theory. A graph

H is called a minor of a graph G if H can be obtained from G by a finite sequence of an

edge contraction or taking a subgraph. We say that a property P of a graph is inherited

by minors if a graph has P then each proper minor of the graph also has P. Let (P) be

the set of all graphs which do not have P and whose all proper minors have P. This set

is called the obstruction set for P and each of the elements in (P) is called a forbidden

graph for P. It is clear that a graph G has P if and only if G does not have a minor

which belongs to (P). Then, according to Robertson-Seymours Graph Minor Theorem

[20], the following fact holds.

Theorem 2.2.1. (Robertson-Seymour [20]) (P) is a finite set.

In particular for the trivializability T, it is known the following.

Proposition 2.2.2. (Taniyama [27]) The trivializability is inherited by minors.

Therefore by Theorem 2.2.1, we have that (T) is a finite set, namely the trivializability

of a graph can be determined by finitely many forbidden graphs. Thus we want to

determine all elements in (T), namely we consider the following problem.

Problem 2.2.3. Find all forbidden graphs for the trivializability.1 Her method can produce a trivializable graph which is not a minor of a bifocal or neo-bifocal. But the author does

not know whether her method can produce a trivializable graph which is not a minor of a 3-line web or not.

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By the works of Sugiura-Suzuki and the author-Ozawa-Taniyama-Tsutsumi, sixteen

forbidden graphs for the trivializability have been found as follows.

Theorem 2.2.4. (1) (Sugiura-Suzuki [23]) The seven graphs G1, G2, . . . , G7 as illus-

trated in Fig. 2.2.1 belong to (T).

(2) (Nikkuni-Ozawa-Taniyama-Tsutsumi [18]) The nine graphs G8, G9, . . . , G16 as illus-

trated in Fig. 2.2.1 belong to (T).

G2 G3G1

G5 G6G4

G7 G9 G10G8

G12 G13G11

G15 G16G14

Fig. 2.2.1. Forbidden graphs G1, G2, . . . , G16 for the trivializability

Indeed, each Gi has a KP fi (i = 1, 2, . . . , 16) as illustrated in Fig. 2.2.2. Actually any

of the spatial embeddings of Gi obtained from fi contains a Hopf link (i = 1, 2, . . . , 16).

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Therefore each Gi is not trivializable. Moreover it can be seen that each of the proper

minors of Gi is also a minor of a 3-line web. Thus by Proposition 2.2.2 we have that

G1, G2, . . . , G16 (T).

Now, we show a new forbidden graph for the trivializability. Let G17 be a planar graphand f17 a regular projection of G17 as illustrated in Fig. 2.2.3. We can see easily that

f17 is a KP, actually we can check that each of the spatial embeddings of G17 obtained

from f17 also contains a Hopf link. Thus G17 is non-trivializable. Moreover we have the

following.

Theorem 2.2.5. The graph G17 belongs to (T).

The proof is the same as the proof for G1, G2, . . . , G16, actually we can see that each

of the proper minors of G17 is a minor of a 3-line web and G17 is not homeomorphic to

G1, G2, . . . , G16. So we omit the details.

It seems that (T) is not determined by these seventeen forbidden graphs. Actually

we have candidates for forbidden graphs for the trivializability. Let H be the graph as

illustrated in Fig. 2.2.4. Sugiura asked the following question in his master thesis [22].

Question 2.2.6. (Sugiura [22]) Is the graph H trivializable?

This question is still open. Since we can see that each of the proper minors of H is

trivializable, ifH is not trivializable then H (T). We remark here that for any regular

projection f of H there exists a spatial embedding f of H obtained from f such that f

does not contain any non-trivial link [18]. Therefore if H is non-trivializable then we can

find a spatial embedding f of H which is obtained from its KP and does not contain any

non-trivial link.

Besides, let H1, H2, H3 and H4 be four graphs as illustrated in Fig. 2.2.5. Then we

have that each Hi has a KP gi (i = 1, 2, 3, 4) as illustrated in Fig. 2.2.6.

Question 2.2.7. (Nikkuni-Ozawa-Taniyama-Tsutsumi [18]) Is each of Hi a forbidden

graph for the trivializability?

Note that each of Hi has a minor which is homeomorphic to H. Thus if H is not

trivializable then H1, H2, H3 and H4 are not forbidden graphs for the trivializability, and

if H is trivializable then there is a possibility that H1, H2, H3, H4 (T).

2.3. Spatial subgraph in a spatial graph obtained from a KP

For the present, any spatial graph obtained from a known KP contains a Hopf link.

Then the following question can be considered in general.

Question 2.3.1. What kind of spatial subgraph does appear in a spatial graph obtained

from a KP?

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8f 9f 10f

11f 12f 13f

14f 15f 16f

7f

4f 5f 6f

1f 2f 3f

Fig. 2.2.2. A KP fi : Gi R2 (i = 1, 2, . . . , 17)

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f

G17

1

2

3

4

5

6

7

8

6

3

8 75

1

4 217

Fig. 2.2.3. New forbidden graph and its KP

H

Fig. 2.2.4.

H2 H3H1 H4

Fig. 2.2.5.

1g 2g 3g 4g

Fig. 2.2.6. A KP gi : Hi R2 (i = 1, 2, 3, 4)

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As a spacial case of Question 2.3.1, the following question was asked in [18].

Question 2.3.2. (Nikkuni-Ozawa-Taniyama-Tsutsumi [18]) Let f be a KP of a planar

graph G and f a spatial embedding of G obtained from f. Does there exist a proper

subgraph H of G such that f|H is a non-trivial spatial embedding?

We remark here that Question 2.3.2 is equivalent to the following question.

Question 2.3.3. (Nikkuni-Ozawa-Taniyama-Tsutsumi [18]) Is a regular projection of a

minimally knotted spatial embedding of a planar graph not a KP?

Here a spatial embedding f of a planar graph G is said to be minimally knotted if f is

non-trivial and f|H is trivial for any proper subgraph H of G. For example, each of the

spatial graphs f, g and h as illustrated in Fig. 2.3.1 is known as a minimally knotted

spatial graph.

In the following we give a partial answer for Question 2.3.3. A spatial embedding f

of a planar graph G is said to be strongly almost trivial [7] if f is non-trivial and there

exists a regular projection f of f such that any of the spatial embeddings of H obtained

from f|H is trivial for any proper subgraph H ofG. At this time we also call f a strongly

almost trivial projection of G. It is clear that if f is strongly almost trivial then it is

minimally knotted. For example, each of the spatial graphs f and g as illustrated in

Fig. 2.3.1 is strongly almost trivial, actually they have f and g as strongly almost trivial

projections, respectively. But a spatial graph h as illustrated in Fig. 2.3.1 is not strongly

almost trivial [9]. Then we have the following.

Proposition 2.3.4. A strongly almost trivial projection of a planar graph is not a KP.

We prepare some definitions and lemmas that are needed to prove Proposition 2.3.4 (We

also need them in the next section). A spatial graph is said to be free if the fundamental

group of the spatial graph complement is a free group. Then the following is Scharlemann-

Thompsons famous criterion.

Lemma 2.3.5. (Scharlemann-Thompson [21]) A spatial embedding of a planar graph is

trivial if and only if any of the spatial subgraphs is free.

On the other hand, the following is known.

Lemma 2.3.6. (Nikkuni-Ozawa-Taniyama-Tsutsumi [18]) Let f be a regular projection

of a graph (does not need to be planar). Then there exists a free spatial embedding of the

graph obtained from f.

Now we give a proof of Proposition 2.3.4. The proof is essentially the same as the proof

of [18, Proposition 3.4]. Let f be a strongly almost trivial spatial embedding of a planar

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f g h

f g h

Fig. 2.3.1.

graph G and f the strongly almost trivial projection. Since f is minimally knotted, by

Lemma 2.3.5 we have that f is not free. On the other hand, by Lemma 2.3.6, there exists

a free spatial embedding g of G obtained from f. Then, by the definition of the strongly

almost triviality, we have that g|H is trivial for any proper subgraph H of G, namely g|His free for any subgraph H of G. Thus by Lemma 2.3.5 we have that g is trivial. So f is

not a KP.

3. Identifiable projection (IP)

3.1. IP and the Simon invariant

When we consider an IP, the following fact is fundamental.

Proposition 3.1.1. (Huh-Taniyama [10]) A non-planar graph does not have an IP.

To explain Proposition 3.1.1, we recall the Wu invariant[30] of a spatial graph. Let X

be a topological space and

C2(X) = {(x, y) X X | x = y}

the configuration space of ordered two points of X. Let be an involution on C2(X)

defined by (x, y) = (y, x). Then we call the integral cohomology group of Ker(1 + #)

the skew-symmetric integral cohomology group of the pair (C2(X), ) and denote it by

H

(C2(X), ). It is known that H2(C2(R3), ) = Z [30]. We denote a generator of

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H2(C2(R3), ) by . Let f be a spatial embedding of a graph G. Then f induces a

homomorphism

(f2) : H2(C2(R3), ) H2(C2(G), ).

We call (f2)() the Wu invariant of f and denote it by L(f). In particular, if G is

homeomorphic to K5 or K3,3 then H2(C2(G), ) = Z [28] and L(f) coincides with the

Simon invariant [26]. We note that L(f) coincides with twice the linking number if G is

homeomorphic to the disjoint union of two 1-spheres. We refer the reader to [28] for a

diagramatic calculation ofL(f).

The generator of H2(C2(R3), ) depends on the orientation of R3 [30], namely it

holds that L(f!) = L(f) for a spatial embedding f of G, where f! denotes the mirror

image off. Since the Simon invariant is odd integer-valued [26], we have that any spatial

embedding of K5 and K3,3 is not ambient isotopic to its mirror image. This implies thatany spatial embedding of a non-planar graph G is not ambient isotopic to its mirror

image because G always contains a subgraph which is homeomorphic to K5 or K3,3 [13].

Therefore a non-planar graph cannot have an IP.

3.2. Characterizing spatial graphs obtained from an IP

Now let us consider an IP f of a planar graph G. If G is trivialzable, then a trivial

spatial embedding of G can be always obtained from f. Hence the following is clear by

the definition of the identifiability and the uniqueness of a trivial spatial embedding.

Proposition 3.2.1. Let G be a trivializable planar graph and f a regular projection

of G. Then, f is an IP if and only if any of the spatial embeddings obtained from the

projection is trivial.

By the existence of a KP, the proof of Proposition 3.2.1 does not work for non-

trivializable graphs as it is. But by an application of Lemmas 2.3.5 and 2.3.6, we can

generalize Proposition 3.2.1 to arbitrary planar graphs as follows.

Theorem 3.2.2. (Nikkuni [16]) A regular projection of a planar graph is an IP if andonly if any of the spatial embeddings obtained from the projection is trivial.

Actually, we have the if part by the uniqueness of the trivial spatial embeddings of

a planar graph. And we have the only if part as follows. Let f be an IP of a planar

graph G and f the spatial embedding of G obtained from f. We note that f|H is also

an IP for any subgraph H of G. Then by Lemma 2.3.6 the spatial embedding g of H

obtained from f|H is free. Since g = f|H, we have that f|H is free for any subgraph H

of G. Therefore by Lemma 2.3.5 we have that f is trivial. This completes the proof of

Theorem 3.2.2.

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A regular projection of a graph is said to be reducedif the image of the projection has

no local parts as illustrated in Fig. 3.2.1 (a) and (b). For example, each of the regular

projections as illustrated in Fig. 1.1.2 (2) and (3) is reduced. Note that there exist

infinitely many planar graphs whose reduced IP are only embeddings from the graph toR

2, see [10] (We also refer the reader to [3], [19], [25], [11], [31] and [8] for related works).

Theorem 3.2.2 indicates that the spatial embedding of a planar graph obtained from an

IP of the graph is always trivial even if the projection is reduced and not an embedding

into R2.

(a) (b)

Fig. 3.2.1.

4. Completely distinguishable projection (CDP)

4.1. CDP of a planar graph

We recall that the knottedness of a regular projection is defined for only planar graphs,

and the identifiability of a regular projection is considered for only planar graphs as

a consequence of Proposition 3.1.1. But the complete distinguishability of a regular

projection can be considered not only for planar graphs but also for non-planar graphs.

First we deal with the case of planar graphs in this subsection. A CDP of a graph is said

to be trivial if it has no double points. Therefore every planar graph has a trivial CDP.

Moreover, a non-trivial CDP of a planar graph is closely related to KP as follows.

Proposition 4.1.1. (1) LetG be a trivializable planar graph. Then a regular projection

f of G is a CDP if and only if it has no double points.(2) LetG be a non-trivializable planar graph. If a regular projectionf ofG is a non-trivial

CDP then f is a KP.

Actually it is clear that any spatial embedding of a graph which can be obtained from a

non-trivial completely distinguishable projection of the graph is not ambient isotopic to

its mirror image. By this fact, we can prove Proposition 4.1.1. We note that the converse

of Proposition 4.1.1 (2) is not true, see Examples 4.1.2 and 4.1.3.

Example 4.1.2. Let G be a non-trivializable planar graph and f a CDP of G. Note

that f is also a KP by Proposition 4.1.1 (2). Then by producing the local parts in f(G)

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as illustrated in Fig. 4.1.1, we can construct another KP which is not a CDP.

Fig. 4.1.1.

Example 4.1.3. Let G be a planar graph and f a regular projection ofG as illustrated

in Fig. 4.1.2. Since f(G) contains a image of a KP of the octahedron graph as illustrated

in Fig 1.1.1, we have that f is a KP. We can also check that f(G) does not contain any

local parts as in Fig. 4.1.1. But Fig. 4.1.3 shows that

f is not a CDP.

f

G

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Fig. 4.1.2.

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Fig. 4.1.3.

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4.2. CDP of a non-planar graph

In this subsection we deal with the case of non-planar graphs. Let Kn be the complete

graph on n vertices and Km,n the complete bipartite graph on m + n vertices. Note that

Kn is non-planar if n 5, and Km,n is non-planar if min{m, n} 3.

Theorem 4.2.1. (Nikkuni [17]) Each of Kn and Km,n has a CDP.

We note that each of Kn for n 4 and Km,n for min{m, n} 2 is planar and trivializ-

able [23]. So in this case only embeddings from the graph intoR2 are CDP by Proposition

4.1.1 (1).

In the following we show that Kn has a CDP. Since Kn has a trivial CDP for n 4, we

assume that n 5. We set V(Kn) = {1, 2, . . . , n}. We denote the edge ofKn connecting

two vertices i and j by ei,j (1 i < j n). Let Wn = Cn1 st(n) be a subgraph of Knembedded in R2 as illustrated in Fig. 4.2.1, where Cn1 = e1,2 e2,3 e1,n1 and

st(n) = n1i=1 ei,n. Then Cn1 divides R2 into two domains and we can construct a regular

projection f : Kn R2 by embedding each of the edges in E(Kn) E(Wn) into the non-

compact domain such that any pair of the edges ei,j and ek,l (1 i < k < j < l n 1)

has exactly one double point and the other pair of the edges has no double points, see Fig.

4.2.1. Then, for the double point P between ei,j and ek,l (1 i < k < j < l n 1),

we can find a subgraph H = Cn1 ei,n ek,n ej,n el,n ei,j ek,l of Kn which is

homeomorphic to K5 such that f|H is a regular projection ofH with exactly one double

point P. Let g and g! be exactly two spatial embeddings of H which can be obtainedfrom f|H. Then we have that |L(g)| = |L(g!)| = 1. Since L(g!) = L(g), we have that

L(g) = L(g!). This implies that f is a CDP.

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61

2 3

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1

2

3 4

5

6

Fig. 4.2.1. A CDP f : Kn R2 (n = 5, 6, 7)

We remark here that the construction above is based on a complete calculation of the

Wu invariants of spatial embeddings of Kn by the author, see [15] for the details.

Conjecture 4.2.2. Every graph has a CDP.

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Since a planar graph always has a trivial CDP, our aim is to prove whether a non-

planar graph always has a CDP. For a planar graph, it is also natural to ask the following

question.

Question 4.2.3. Does a non-trivializable planar graph have a non-trivial CDP?

4.3. CDP and the minimal crossing projection

Let cr(f) be the number of double points of a regular projection f : G R2. Then we

call cr(G) = min{cr(f) | f : G R2 is a regular projection} a minimal crossing number

ofG. A regular projection f ofG is called a minimal crossing projectionifcr(f) = cr(G).

For example, cr(K5) = cr(K3,3) = 1, cr(K3,4) = 2 and cr(K6) = 3 (cf. [6]). Fig. 4.3.1

shows an example of the minimal crossing projections of K5, K3,3, K3,4 and K6.

Fig. 4.3.1. Minimal crossing projections

Let f be a CDP ofKn (resp. Km,n) constructed by the method in the proof of Theorem

4.2.1. Then it can be easily seen that f is not a minimal crossing projection if n 6

(resp. min{m, n} 3 and max{m, n} 4). On the other hand, we can check that each

of the minimal crossing projections as illustrated in Fig. 4.3.1 is a CDP in a similar

way as the proof of Theorem 4.2.1. From this viewpoint, Ozawa suggested the following

question.

Question 4.3.1. ([17]) Is a minimal crossing projection a CDP?

Note that a minimal crossing projection of a planar graph is an embedding of the graphinto R2, namely a trivial CDP. Besides we can give an affirmative answer to Question

4.3.1 for the case of minimal crossing number one [17].

We present a piece of circumstantial evidence as follows.

Example 4.3.2. Let f be the regular projection of K2n, which is Blazek-Komans

construction of a candidate for the minimal crossing projection of K2n [1]. Fig. 4.3.2

illustrates the case of n = 4 and in this case f is a minimal crossing projection of K8.2

2 R. K. Guy conjectured in [4] that

cr(Kn) =1

4

n

2

n 1

2

n 2

2

n 3

2

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Then we can see that f is a CDP in a similar way as the proof of Theorem 4.2.1.

1

2 3

4

5

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1

2 3

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6 7

8

Fig. 4.3.2. f : K2n R2 (n = 4)

Example 4.3.3. Let f be the regular projection of Km,n, which is Zarankiewiczs con-

struction of a candidate for the minimal crossing projection of Km,n [32]. Fig. 4.3.3

illustrates the case of m = n = 6 and in this case f is a minimal crossing projection of

K6,6.3 Then we can also see that f is a CDP in a similar way as the proof of Theorem

4.2.1.

1

3

5

7

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11

2 4 6 8 10 12

1

3

5

7

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11

2 4 6 8 10 12

Fig. 4.3.3. f : Km,n R2 (m = n = 6)

Conjecture 4.3.4. A minimal crossing projection is a CDP.

We remark here that if Conjecture 4.3.4 is solved affirmatively then Conjecture 4.2.2

is also solved affirmatively.

and showed that the conjecture above is true for n 10 [5].3 From the beginning, the crossing number problem for Km,n is called Tur ans brick factory problem (cf. [29]). In [32],

K. Zarankiewicz claimed that

cr(Km,n) = m

2

m 1

2

n

2

n 1

2

.

However Guy pointed out a gap for Zarankiewiczs proof [4], and now this is called Zarankiewiczs conjecture. At present,

Zarankiewiczs conjecture is true for m 6 [12].

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Department of Mathematics, Faculty of Education, Kanazawa University

Kakuma-machi, Kanazawa, 920-1192, Japan

[email protected]

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ryo nikkuni- regular projections of spatial graphs

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