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Xerox University Microfilms 300 North Zeeb Road Ann Arbor, Michigan 4B106 I I I I 75-26,683 WANG, Chin San, 1939- IMBEDDING GRAPHS IN THE PROJECTIVE PLANE.
The Ohio State University, Ph.D., 1975 Mathematics
Xerox University Microfilms ,Ann Arbor, Michigan 48106
THIS DISSERTATION HAS BEEN MICROFILMED EXACTLY AS RECEIVED. EMBEDDING GRAPHS IN THE PROJECTIVE PLANE
DISSERTATION
Presented in Partial Fulfillment of the Requirements for
the Degree Doctor of Philosophy in the Graduate
School o f The Ohio S ta te U n iv ersity
Chin San Wang, B .S ., M.S.
* * * #
The Ohio State University
1975
Reading Committee: Approved Ry
G. Neil Robertson
Henry H. Glover
Department of Mathematics ACKNOWLEDGEMENTS
I wish to thank my advisers Henry Glover and Philip Huneke for giving me this thesis problem; Professor Huneke for his care ful detailed supervision of my computations, Professor Glover for his general understanding of this area.
I particularly appreciate the time that we spent together checking my work ( including many weekends and holidays. )
i i VITA.
December 18, 1939 ••• Bora - Ilan, Taiwan
1958 ...... Hwa Lien Normal School, Hwa Lien, Taiwan
1963 ...... Taichung Teachers College, Taichung, Taiwan
1967 - 1968 ...... Mathematics Instructor, Brashear High School Brashear, Missouri
1968 ...... B.S. in Mathematics, Northeast Missouri State University, Kirksville, Missouri
1968 - 1970 ...... Teaching Assistant, University of Arkansas Fayetteville, Arkansas
197O ...... M.S. in Mathematics, University of Arkansas Fayetteville, Arkansas
1970 - 1975 ...... Teaching Associate, The Ohio State University Columbus, Ohio
1 9 7 5 ...... M.S. in Computer and Information Science Ph.D. in Mathematics, The Ohio State University Columbus, Ohio (anticipated)
FIELDS OF STUDY
Algebra : Erofessor Joseph C. Ferrar
Analysis : Erofessor Bogdan Baishanski
Topology : Professor Henry H. Glover
i i i TABLE OF CONTENTS
Page ACKNOWLEDGMENTS...... i i
VITA ...... H i
INTRODUCTION ...... 1
SECTION
1 . THE PARTIALLY ORDERED SET l ( M ) ...... 5
2 . SOURCES, SINKS AND SYMMETRIES IN l( M ) ...... 8
3 . THE TOPOLOGY OF THE PROJECTIVE PLANE...... 10
k. DISTINCTNESS OF GRAPHS IN l ( P ) ...... 15
5* THE MAIN RESULT ...... 27
6 . SOURCES DO NOT EMBED...... 29
7* SINKS ARE IRREDUCIBLE ...... k2
8 o SOURCES AND SINKS OF A GRAPH ...... 67
9« CONJECTURES AND RELATED QUESTIONS...... 73
APPENDIX...... 75
BIBLIOGRAPHY...... I l l
INDEX OF GRAPHS IN l ( P ) ...... 112
iv INTRODUCTION
The four color conjecture is well knowno In 1 8 9 0 p . j .
Heawood [ 5 ] fcund the mistake in Kempe's "proof" of this
conjecture and proved the corresponding result for five colors*
In addition he formulated a generalization of the four color
conjecture to general surfaces* He conjectured that the chromatic
number n , the number of colors needed to color the closed
orientable surface of genus g , M is the integer part C&
and for the closed non-orientable surface of genus g t M i s O
The Heawood conjecture for non-orient able surfaces was
solved by Ring el [ 9 1 in 1952 and by Ringel and Youngs [10] for
orientable surfaces in 1968* They used Heawood's "contiguity
Lemma” which allowed them to replace th is problem with an embedding problem* Namely they showed that the orientable genus, the
smallest genus of any orientable surface containing is the integer hull
1 2
(n - k)(n - 3 ) g(n) 12 and the ncn-orientable genus of is
(n - 4Kn - 3) g(n)
The idea used in this replacement is an amalgamation technique similar to that in the proof of the five color theorem. Using this technique ve may construct an example of a map on M (*L) with D O exactly n(g) (rT(g)) countries each two of which are contiguous.
By considering the dual to the graph of boundaries of this map (the graph of railroads between the capitals of these countries) we then get c Mg (K^ c Mg) and the dual embedding problem (see
Ring el [ 8 ]) •
Using the solution of the Heawood conjecture as motivation we next ask whether we can find a method to compute the orientable
(respectively non-orient able) genus of an arbitrary graph. That is, given arbitrary graph K find the integer g such that M 6 contains K but M ... does not contain K (respectively M & “* o contains K but M_ m, does not contain K). Q An answer to a special case of this question is the theorem 2 of KuratowskL [ 7 ] which says that a graph is planar (K c IR ) if and only if K does not contain or ^ • Equivalently, g(K) - 0 if and only if K does nob contain or ^ • 3
A first generalization of KuratowskL 's theorem would be a
similar characterization of graphs that embed in the projective p lan e P •
In this paper we prove part of this characterization. In particular we prove that the set of irreducible graphs l(P) for p P (analogous to the set l( R ) consisting of and contains 103 graphs (Theorem 5*1)•
Further we strongly conjecture that our list of 103 graphs gives the complete set l(P) • (See section 9 for a more complete discussion of this conjecture.)
The technique that we use to construct our lis t of 103 graphs in l(p) is as follows: Given a surface M , Paul Kainen [ 6 ] introduced a partial ordering on l(M) , the set of irreducible graphs for M • Huneke and Glover [ 2 ] have introduced a study of the isotcpy classes of embeddings of K c M which makes it possible to study this partially ordered set. In particular they strongly conjecture that l(P) has exactly five maximal elements.
One of these five graphs was found by Hell Robertson. The 103 graphs we have found are obtained by using this partial order to construct graphs below these five. In this paper we do not show that these are all graphs K in l(p) below these five, but we do prove that each of the 103 graphs given are distinct and are in l(p) • The proof that these five graphs are the maximal elements will appear in [ 3 ]• The proof that these 103 graphs are all the graphs in l(P) below these five will appear in [11]. Altogether this will show that l(P) Is precisely the set of these 103 graphs.
The plan of the paper follows: In section 1 we give basic definitions and prove that l(P) is a partially ordered set with respect to the order that we introduce (Preposition 1.2). In
se ctio n 2 we discuss certain special elements in l(p) which are
sinks and sources of a partially ordered subset of l(P) • Many of these sources (minimal minors) were found independently by
Robertson. We also discuss symmetries in graphs. In section 3 we give miscellaneous facts about embedding graphs in the projective plane P • In section 4 we prove that all 103 graphs we lis t are distinct (not homeomorphic)* In section 3 we use results in sections 6 - 8 to prove our main result: Theorem 5.1 - l(p) contains 103 graphs. In section 6 we prove that sources do not embed in P • In section 7 we prove that sinks are irreducible*
In section 8 we complete the proof of Theorem 3.1 by locating each graph between a source and a sink* In section 9 we discuss conjectures and unsolved problems related to our results. An index is included giving the pages on which each of the 103 graphs are discussed. 1 . THE PARTIALLY ORDERED SET l(M )
By a surface M we mean a 2-manifold, that is a locally
Euclidean Hausdorff space of dimension 2 •
By a graph K we mean a finite 1-dimensional CW complex.
By an embedding C: K ■+■ M we mean an injective map (continuous
fu n c tio n ). When we w rite K c M o r M ^ K we mean t h a t th e re
exists an embedding C : K ■+• M •
Given a surface M we say that a graph K is irreducible
for M If K ^ M but for every proper subspace L 5K , L c M.
Given surface M let l(M) denote the set of all Irreducible graphs for M. As an example we note that Kuratowski's theorem
says th a t
l(iR2) = {K^K^}
Given surface M we next Introduce a relation on l(M) • As motivation we Observe that ^ may be obtained from by
splitting any vertex and deleting two edges.
Figure 1.1 5 6
Specifically, given graph K we may form a new graph K
constructed by replacing a vertex v € K by two vertices v 1 and
Vg connected by an edge e and then connecting seme of the
edges Incident to v In K so that they beccme Incident to
v^ € k and the remaining edges In K incident to v so that they r become Incident to Vg .
To describe the operation above, let adj(v) denote the
set of all vertices In K adjacent to v . Suppose
adj(v) = (u^, u2, ..., u q) • If K is the graph resulting from
splitting v Into vir Vg with adjCv^) = Ug, ..*, ujj) In
K, then we write K = § / , \K • When referring to an v:^u^, •••, u ^
arbitrary splitting of v , we shall use the notation S^(K) .
Furthermore, when the underlying graph K is obvious,
v: (i^, ..., u^ Is used Instead of ^ ^ )K.
K - e w ill denote the graph given by deleting an edge, more particularly the interior of an edge* K/e w ill denote the graph given by contracting e (collapsing e into a point)*
K - st(v) will denote the operation of deleting the star of a vertex, i*e* v together with the interiors of all incident edges*
These operations may involve a set of edges or vertices instead
of a single edge or vertex*
Lemma 1*1 If K M, then
Proof: Assume <^K CM, i.e. there exists embedding
f: <^K M. Note that there exists a commutative diagram 7
f * M
f y sTI^ e W t W
H i K * M
where f and f are induced by f and are hence embeddings*
Let K € l(M) , since ^(k) ^ M, there exists at least
one graph L c ^(K) such that L € i(m) • We define each
such L to be less than K and define a relation < on
l(M) to be the transitive relation generated by the above*
This means that if L < K and J < L then J < K.
Proposition 1*2 The relation < cm l(M) is a partial ordering*
Proof: We need only check anti**symmetry* Observe that the
lexicographical ordering of the valency sequences (each
written as a non-increasing sequence) of graphs K € i (m)
gives a partial ordering on l(M) which refines < on
l(M) • Hence the result* 2 . SOURCES^ SINKS AND SYMMETRIES IN l(M )
We first discuss sources and sinks*
K € l(M) is called a source if K/e CM for all edges e
in K. K £ l(M) is called a sink if ^.K is reducible for
every vertex v of K •
Lemma 2.1 Let L = , K £ M, L €i (m). Then K € l(M) .
Proof: K » l/e for some edge e € L • Thus
K - e' = L - e'/e* Since L - e' CM , so does L - e'/e •
T.prmrift P.P Let K1 = ^K 1 ’ 1 , i » 1, 2, *.*, n. If K°
and K? €i(m) , then K1 € i(m) for all i » 1, 2, n .
The proof of Lemma 2*2 follows immediately from 2*1 •
We next discuss symmetry in graphs* If K is a simplicial graph without vertices of valency 2 , then every homaomorphism of K induces a permutation of the vertices of K • Note that every such permutation is a product of (simple) cycles* A hameomoxphism which induces a permutation on vertices is called a symmetry on K • The notation
S : C(vx , v2 , vk ), ... , v2 , ..., vp )]
8 ■wil l represent a hameamorphism C which induces the permutation
given hy the product of the given cycles. Also we write v^, «v
i f v^ , Vj are in the same cycle of s • We also write
e ~ e* if e = (v-^Vg) , e' « (v-^Vg1) and S ^ ) » v^ ,
S(v2) «» Vg1 • The S will be deleted and we will write e ~ e 1 v ~ v 1 when S is clear from the context* 3* THE TOPOLOGY OF THE PROJECTIVE PLANE
Recall that the real projective plane P is defined to be 2 the orbit space of the antipodal involution on the 2-sphere S • 2 Note that P is home amorphic to the unit disc in JR with
antipodal points in the boundary identified, a simple cycle C
in P is called essential if P - C is connected; null if
P - C is not connected.
For a graph K > (v^,Vg) denotes an edge with incident vertices v^ vg . Further, (v^ Vg, vQ+1) will denote a path from v^ to vn + ^ • If vi a vn + 1 "then we say
(v^ ..., v ) is a n-cycle.
A graph K is called projective if there exists an
embedding C: K P • K is non-pro j active otherwise. A
cycle C in K is said to be essential with respect to an
embedding £ if £(c) is essential in P • A cycle C in K
is said to be essential for P if for every embedding of
£: K P , C is essential with respect to £ •
A subgraph L of a graph K is called a fc-graph if there
exists a graph M such that L c M c K and either
(i) L is homaomorphic to Kg ^ , and M is homeomorphic to
jc „ with one of the cubic vertices of M not in L
10 11
o r
(ii) L Is home amorphic to , and M is hctmeomorphic
t o K^.j o or r S with the cubic vertices of M not in L • a / a b x We use the notation ^ c d j to represent K
/ £t b \ a £» b t ^ I x y J re p re se n t x y z
In case a k-graph L in K contains vertices of K which
are valency 2 in L , the notation w ill be expanded to include
those vertices. For example
a b , re p re se n ts 2 C\ x y z ' z
For most of the 10J graphs, each contains a subgraph homemorphic to ^ . To study the (non) embedding of these graphs we often embed this ^ into P • As an example the graph below represents a graph K in which ( 2 ,b ) , ( 2 ,a), (c,l) are
edges in K (through points of identification in P )•
F ig u re 5 .1 We close this section by giving the following lemmas which will be used in later sections. 12
Lgmmft. ^.l if a graph K 'with £ edges and V vertices embeds in P with F faces (components of the complement of K in O P ) , each of which Is home amorphic in 1 , then V ■ E + F =* 1 •
Proof: The Euler characteristic of P is I t The given
vertices, edges, and faces give a CW decomposition of p •
Hence the result*
Lemma 5.2 Any two essential cycles and Cg in P intersect each other.
Proof: We use that, for essential, P - is 2 hamaomorphic to HR • This is a well known fact of
topology. Assume Cg is disjoint from • By the 2 Jordan curve theorem B - Cg has 2 components.
Hence P - C^ - Cg has two components and C^ is in
the boundary of only one of these components. Hence
P - Cg has 2 components. Hence Cg is not essential.
Lemma 5.5 Let K be a graph and let C: K + P be an embedding.
Suppose Ci are cycles in K, i « l , ...,n such th a t d - i (i) C. H U C. is nonempty and connected for each J i =1
j « 2, •.•, n , and
(li) C^ is null with respect to C for each i « 1 , . . . , n .
n Then C ( U C .) is n u l l, i «1 13 n P ro o f; I f U C. contains a cycle C which is essential i = l ~ 2 n with respect to C th en P - CC ■ E • Since U C. i «1 1 n is connected, each component of P - C( U C .) i s i =1 p n homaomorphic to HR and P - U C. has 1 - V + E i =1 1
components by Lemma J .l • However, n , the number of n independent cycles in U c. , is 1-V + E, the Betti i =1 1 n number o f U C. • It remains to prove by induction on i =1 1 k k that P - C ( U C.) has at least k + 1 components i =1
and hence F > n + 1 , a contradiction* For k = 1 ,
Cc± is null so that P - C( C^) has two components*
For each k , P - £(C.) has two components, each of * k-1 which intersect one of the components of P - C ( U C .) • i *1 x k “ 1 Hence, assuming inductively that P - C ( U C.) has a t i =1 k least k components, P - C( U C.) has at least k+ 1 i =1
components* Hence the result*
T,amnia. Let K be non-projective, let (v1,v2,v^) be a
3“cycle in K with v^ cubic (valency 3) then K - (v2,Vj) is non-proj active• 11*
Proof: Suppose there is an embedding C: K - (v_,v_) P •
Use a neighborhood of the edges (v^Vg) and to
give an embedding of K • This contradiction proves the re s u lt*
Lemma 5.5 If L is a k-graph in K and C: K P is an embedding, then there is a cycle C c L such that C(c) i s essential in P .
Proof: it is sufficient to assume that K = or ^ •
Suppose C(C) is null for all C 01 L • Then th e re
exists Cq c P disjoint from C(L) which is essential*
As a result L CP - CQ - E 2 such that L(1(K-L)
is contained in the closure of 1 region. Since K has O only one vertex not in L , K c 3R contradicting
Kuratowski's theorem* Hence the result* k. DISTINCTNESS OF GRAPHS IN l( P )
We shall prove that the 103 graphs in the appendix are pairwise distinct*
We name our graphs according to their first Betti number
(3 = 1- V + E where V, E denote the number of vertices and edges of the graph respectively* More specifically each A^ is a graph with Betti number p » 12, B^ with p =» 11, with
P alO, D^ w ith p = 9 t ^ w ith p =* 8 , w ith p = 7 , and
G is the only graph with p - 6 . Let us call all A^'s the
A-graphs, and similarly for B^*s, Q^'s, etc. Since the Betti number p is clearly invariant under graph isomorphism, we see all A-graphs are distinct from B-graphs, and vice versa* Similarly for C (D, E> F, G) - graph s.
The valency sequences provided under each graph in the appendix are in the general form sQ (s^, Sg, *.., s^) where Sq =* n denotes the number of vertices of the graphs and si (i > 0 ) i s the valency of each of the vertices arranged in descending order.
I f L, M a re graphs of th e same c la s s (e .g *, both are A-graphs) with sequences sQ (s^, Sg, ..*, &Q) , and tQ (t^, tg, ..*, tn) th en L, M are isomorphic implies that = t^ for all i a 0, 1, ..., n . Denote [sQ (s^, Sg, ••*, sq)] » (G1 such that
i
15 16 the valency sequence of G is s^ (s^# s • •*, b^))• We need only to check that the graphs in [Sq ( , fig# ••• > 8n) 3 are distinct from each other*
Looking into the appendix, we see for A-graphs, C-graphs and G-graphs each [sQ (s^* • ••* sn)3 is a singlton set, hence each is distinct from each other* Thus all we have left to check are the classes [ sq ( s^ , •*., s&)] in B-graphs, D-graphs, E-graphs and E-graphs*
Theorem 4.1 The 103 graphs in the appendix are distinct.
Proof: The format of our proof is as follows:
For each class we first list [sQ (s^, ..., sn)3 "by
lexicographic order. Then for each subclass we exhibit a
binary tree which w ill distinguish each graph in the same
class* Each edge in the binary tree is distinguished by a
property or its logical complement P^ • The property
P^ will be stated precisely following the binary tree* As
a hypothetical example consider
[s 0 ( s ^ s 2 , * * *, 1 •••» Ani, )
A. A. n. Figure ^-1 ‘3 17
P^: K I s a disconnected graph*
P,,: The 3 vertices of valency 4 are in a 3“cycle in K.
P..: K contains no five cycle.
In the discussion that follows will be omitted as it is obvious from the binary tree when P^ is specified*
Let K be a graph, U c V(K) , recall the definition from section 1 that K - U denotes the graph resulting by deleting each vertex in U and its incident edges* The following notations w ill be used frequently*
V(Si# sg, s^) a {v €v ( k ) | v is a vertex of valency
s^ f o r some i ■ 1 , 2 , .* , p
V(s1# s2, *.., sp) « Cv €v(K) | v €v(sx, .*., sp) or thereN e x is ts u 6 v (s 1# •••# sp ) I
such that v is adjacent /
t o u \
Note that 7 is defined in similar fashion from V •
4-1 B-graphs
The following are those non-singleton classes
(1) [8 (6 , 6 , 4, 4, 4, 4, 4, 4)] « [Bg, Bg, B^) 18
B,
Figure 4-2
P^: K has vertex-connectivity 2 •
Pgi Each vertices of valency less than 6 in K is adjacent
to a vertice of valency 6 •
(2) [9 (6 , It-, k, k, k, It-, It, k, It-)] =. {Bj, B10)
B3
Figure 4-3
P: K has vertex-connectivity 2*
(3 ) [10(4* 4* 4* 4* 4* 4* 4* 4* 4* 4 )] ™ (Bx , 2 B.j 1 }
* 1
1 1
F igure 4-4 19
P^: K Is disconnected.
Pgj K has vertex-connectivity 2 •
4-2 D-graphs
The following are non-singleton class on valency sequences*
(1 ) [9(5* 5# 4 , 4 , 3* 3* 3 . 3)3 = i \ , Dy Py
12
F igure 4-5
P1: The two vertices of valency 5 are adjacent in K.
Pg: The vertices of valency 4 form a 5-cycle in K.
P..: K - v(5,4) is a 4-cycle. 5
( 2 ) [10(5, 4, 4, 4, 4, 5, 3, 3, 3, 3)3 - 1%, 3>g, D ^}
F igure 4-6 20
P1: K - V(5,3) is a
Pgj K - V(5>3) contains a 3 -c y c le .
(3) [10(5, 5, 4, 4, 3, 3, 3, 3, 3, 3)3 =■ O y ^ D ^}
20
F ig u re 4-7
P1: K contains no 3 -c y c le .
Pgj K - V(4) contains an isolated vertex.
(4) [11(4, 4, 4, 4, 4, 3, 3, 3, 3, 3, 3)3 » (D10, D^,
F ig u re 4-8
P^: K is disccnnected
Pg? K has verfcex-connectivity 2
4-5 E-graphs
Only the following are non-singleton classes. 21
(1) [11(6, 5, 3, 3, 3, 3, 3, 3, 3, 3, 3)3 - Ce^ Egg}
Figure 4-9
P: K is point-separated
(2) C9(5, 5, 3, 3, 3, 3, 3, 3)1 - (E^, E^}
*U
Figure 4-10
P: V(5,4) are mutually non-adjacent in K .
(3) [10(5, 5, 3, 3, 3, 3, 3, 3, 3, 3)3 “ C^, E^, E^}
E.,
Figure 4-11
P^: K - V(5) is a single vertex.
P2: K - V(5) consists of 2 isolated vertices. 22
(4 ) [ 1 0 (5 # 4 , 4 , 3 , 3 1 3 # 3, 3, 3, 3)3 a [^ 5 *
E-
W Figure 4-12 P^: v(4) in K are mutually non-adjacent.
P2: K - V(5,4) is a single vertex
P^: K - V(4) = J
(5 ) [11(5, 4, 3 , 3,3, 3, 3, 3, 3, 3,3)3 - O y E^, E^g)
*32
Figure 4-13
P^: V(5,4) are adjacent in K
Pgi K is not 3 -connected
(6 ) [12(5, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3)3 - (Eg, E^}
Figure 4-14
P: K is point-separated (7) C9(4, 4, 4, 4, 4, 3* 3 1 3 f 3)3 = (Ej,, Eg, Egi> 23
'21 Figure 4-15
P^: K - V(4) is a 4-cycle* P2: K - V(4) = Y P,: K - V(4) is a graph of 4 isolated vertices* 3
( 8) [10(4, 4, 4, 4, 3 , 3, 3, 3, 3, 3)3-0^ E^, E&y E ^, ^ 8 ' E40*
22 '40 Figure 4-16
Px: K - V(3) is P2 : K - V(3) A P-: K - V(4) is a single vertex* 3 P^: K - V(3) is a graph of 4 isolated points.
P_: K - V(3) is a 4-cycle. 5
Cll<4,4,4*3, 3,3,3,3,3,3,3)] - \ k> *^Q>
Figure 4-17 > w 1 is a graph of 3 isolated vertices. pl : K - V(4) c o n ta in s 2 isolated vertices. P2 :
K - V W is a single isolated vertex. P3 : V K - V(4) is a graph of two isolated vertices V K - V(4) are in a 4-cycle in K.
P*: K - V(4) is a null graph.
[12(4j4>3>3j3j3j3/3j3>3*3*3)3 0 ^ 1 9 * V V
Figure 4-18 25
p^: V(^) are adjacent in K.
Pgt K - v(4) is a graph of 3 isolated vertices.
P,: K - V(4) is disconnected. 5
(11) [13(^3,3,3,3,3,3,3,3,3,3,3,3)3 = C*y
. Figure 4--19
P: K is point-separated.
4-^ F-graphs
The following are non-singleton classes.
(1) [10(M ,3,3,3,3,3,3,3,3)3 = [Fg, Fy Fk, Fy Pg )
F igure k-2.0
P^: V(if) are adjacent in K.
Pgj K - V(4) is disconnected. 26
P^: K - V(4) a
P^: K - V(4) > 2 isolated vertices with distance 2 (the
length of the shortest path from one to the
o th e r ) •
(2) [11(4,3,3,5,3,3,3,5,3,3,3)] = (P?, Pg, F9, F10)
10
Figure 4-21
P^: K - V(4) is disconnected.
Pgi K - V(4) is a null graph.
K is not 3-connected.
(3) [12(3,3,3,3,3,3,3,3,3,3,3,3)] - {Fn , P12, P^, F^, F15} These are five of the six distinct cubic irreducible graphs found by Glover and Huneke [ 4] . 5« THE MAIN RESULT
We outline the results of sections 6, 7, Q and from these prove our main result that each of the 103 graphs of the appendix are In l(P)j these 103 graphs are distinct by section 4. Hence:
Theorem 5.1 l(P) contains 103 graphs.
Recall by Lemmas 1.1 and 2.1: if L ** SyK, then K non- projective implies L non-projective. Furthermore L irreducible implies K irreducible. In particular if K1 = and
K2 a , and Y? is non-projective then K 2 , are non- pro J active. If furthermore, is irreducible then K 2 , ¥? are irreducible, and hence K** is in l(P) for i = 1, 2, 3 •
We will prove that each source does not embed in P in se c tio n 6 and that each sink is irreducible in section 7 * In view of the Lemma 2 =2 , for graphs which are not both a source and a sink, we shall consider a directed graph structure in l(p) as follows: Each element in l(p) is a vertex and there is an edge from K to L if L = • We include in section
8 a spanning sub-directed graph of the directed graph l(P) in which every graph in l(P) follows a source and is followed by a sink. More precisely, we observe the following:
27 28
(1) If K Is both a source and a sink then the non-
projectivity and the irreducibility of K are discussed
in section 6 and section 7 respectively*
(2) If K is a source but not a sink then non-projectivity
of K is discussed in section 6. To see irreducibility
one finds a branch of the spanning sub-graph of l(P)
in section 8
K -► K1 -► . . . K?
with kP a sink. Since irreducibility of K is
implied by that of K 11 one needs only to check
irreducibility of K ?3 in section 7 *
(3) Similarly , for a graph which is a sink but not a source
its irreducibility is checked in 7 and non-projectivity
follows from section 8 and section 6.
(4) If K is neither a source nor a sink, then to see that
K is in l(P) one finds a path in section 8
-*■ ••• K . . .
such that K1 is a source and K*1 is a sink. Hence
K is in l(P) by Lemma 2.2
This discussion with the results of sections 6, 7 and 8 completes a proof of our main result.
Page references for the discussion of each graph is given in th e Index. 6 . SOURCES DO NOT EMBED
There are, among the 103 graphs, 35 sources* Each is proved
to be nan-projective* Each source w ill be considered as a
sep arate case M i, 1 < i < 35 •
Lemma 6.1 If K is a spanning subgraph of the complete bipartite graph Kn m with more than 2 (n + m - l) edges, then
K is non-projective*
Proof: Suppose K embeds in P . Then by Lemma 3*1*
l = V- E + F<(n + m) - 2(n + m - l) + F , so F>n + m- l.
However, the boundary of each face contains a cycle and each
cycle in m, hence in K, contains at least four edges* Also each edge is in the boundary of at most two faces, so
2E > 4f , hence n + m - l>F, a contradiction. Therefore,
K does no t embed in P •
Case (Ml) , graph E ^0 •
F igure 6 .1
Observe that E^q = Kj ^ and has 13 edges • Hence E^0 i s
29 30
non-projective by Lemma 6.1 •
Case CM 2) , graph E ^0 .
F igure 6 .2
Observe that is a subgraph of ^ with 15 edges so
it is non-projactive by Lemma 6.1 •
Lemma 6.2 Let K be a graph which embeds in P . Let C be
a cycle in K disjoint from a k-graph of K • Then C is null
in P .
Proof: By Lemma 5*5 for any embedding of K in P , there
is a cycle in each k-graph which is essential. By Lemma 5*2
P contains no disjoint essential cycles so C is null.
Lemma 6.5 Let K be a graph containing cycles ,
i =» 1 , ..., n such that: j - 1 (l) C. H U c. is nonempty and connected f o r each J i =1
j =» 2 , ...,n , and
(2) Ci is disjoint from a k-graph in K for each
i a 1 , ..., n , and n (3 ) ^ C. contains a k-graph in K • i =>1
Then K is non-projective. 31
Proof: Assume K c P • By Lemma 5»2 Is null for each n 1 ® 1, • •*, n • Hence by Lemma 3*3 U C. Is null* i ®1 n However, by Lemma 3*3 U C, must be essential, a i * 1 1 contradiction* Hence the result*
Case (M3) , graph E^ .
F igure 6.3 The cycles = (l,a,3*b) , Cg = (a,2,y,3) , C^ - (3*y>ac,b) satisfy the hypothesis of Lemma 6*3 as
(x 3 z)* ) c ^ = 1 is a k-graph in E^ • Hence is non-projective.
Case (M4) , graph •
F ig u re 6.H-
The cycles « (l,Pjq.) , Cg ® (2,p,q) , C^ - (c,p,q) satisfy the hypothesis of Lemma 6*3 as (,v>) • is a k-graph. Hence is non-projective.
Case (M_5) , graph Fx .
F igure 6*5
The cycles » (1jP>4>x) y Cg « (a,p,a,x) satisfy
Lemma 6 .3 a s
n \ 1 x a / is a k-graph in F^ • Hence F^ is nan-projective.
Case (m 6) , graph B- .
F igure 6 .6
The cycles C.^ = (i/P/o) for i =* 1, 2, 3 satisfy Lemma 6*3 as (iV,) = ^ is a k-graph in • Hence is non-projective.
Case (M7) , graph . 53
F ig u re 6.7
The cycles = (i,i+l,i+2) for i ® 1, 2, 3 and
=» (1,3>5) satisfy Lemma 6.3 as
k d c k * is a k-graph in A^ • Hence is non-projective.
Lemma 6.4 If K is nan-projective, adj(v) = Cvi>v 2 * v3t^ v, v^, ..., v^ 6 V(K) • Suppose v-^Vg a re a d ja c e n t, th en
Svj(v 1 ,v2)(K) “ (vl 'v2 ^ is non“Pr°Jectiv®*
Proof: Let v' he the new vertex of )(K) - (v^Vg) •
v* is cubic and cycle (v’,v 1 ,vg) is a 3“cycle. Hence
the result by Lemma 3*^ •
Case (m8) , graph B^g .
F ig u re 6 .8
Using the notation of Case (M7) j ** ^2*(^,6)A4 "
and A^_ is non*projective by Case (M7) t so B^g is non- 34 projective by Lemma 6.4 •
Case (M.9) , graph .
F igure 6 .9
Using the notation of Case (M 8) , C^ »
So by Lemma 6.4, CQ is nan-projective. y
Case (M10) , graph Cg . ,
Figure 6.10
Using the notation of Case (m 8) , C8 " *2: (1,3) ^12^ " * So Cg is nan-projective by Lemma 6.4 •
C a s S -im jJ > graph .
Figure 6.11
Using the notation of Case (M10) , = ^jc*(l,p)^C 8 ^ " # So IL_ is non-projective by Lemma 6.4 • 35
Case (M12) , graph E^g .
Figure 6.12
Using the notation of Case (Mil) , Ey,g = *y»(p 2)^7^ " * So Egg is non-projective by Lemma 6.4 .
Case (M13_) , graph E ^ .
Figure 6.15
Using the notation of Case (Mk) , Eg^ =* ^ ’(p ,^ 13!^ " * So E~- is non^projective by Lemma 6.k
Lemma 6.5 If K contains two disjoint k-graphs, then K is non-projective•
Proof? Let L be a k-graph in K which is disjoint from
a k-graph L' in K • By Lemma 3*5 L and L' each contain
a cycle which is essential with respect to any embedding
of K. However L H L' = $ and P contains no disjoint
essential cycles by Lemma 3*2 • Hence the result.
In each case (Mi), lk < i < 19 / each graph either is
disconnected, each component containing a k-graph, or contains a 56 outpoint whose complement contains two components, each containing a k-graph of K • Hence the graphs in these cases are non- projective by Lemma 6 .5 .
Case CM 14) , graph V .
Case (M15) , graph JJ.1^ .
Case (M l6) , graph C^ =* v ^ .
Case (M17) , graph D^g = K^J_LK^ ^ •
Case (M l8 ) , graph => V .
Case (M19) , graph F15 = K3,^'LLlb ,3 ’
In each of the remaining cases (Mi) 20 < i < 35 * either the graph contains disjoint k-graphs , so it is non-projective by
Lemma 6.5 (the disjoint k-graphs are listed in each of these cases) or the graph is ncn-projactive by Lemma 6.4 (the splitting is given in each of these cases).
Case (M 20) , graph c1ft.
Figure 6.14
are disjoint k-graphs. Case (m 21) , graph D-^
Figure 6.15 G1 2D ^ and b ^ are disjoint k-graphs.
Case CM 22) , graph Dg 0
F igure 6.16
/ a b \ / c d \ \ 1 2 3 / 031(1 \ 4 5 6 / are disjoint k-graphs.
Case (M23) , graph
Figure 6.17
( & X \ f ( ^ are d is jo in t k-graphs. Vp q./ Vl 2 3 / Case (M 24) , graph E,
F ig u re 6. l 8
E^0 » ^(p^C^-Lg) “ (p>q), see Case (M2?) for notation
Case (K25)
Figure 6.19
®59 = ^c:(p,q)^Ei 6^ " ^P,q^ ' seQ Case (M25^ for nofcaticn fiage.„.(M £6); graph D12 *
Figure 6.20 r xwb c)a re 2 disjoint k-graphs. \ P 1 ' v i 2 i J
Case (M2?) t graph E^ . 59
Figure 6.21
E4 * Sx*(p q)^D 12^ " * see 0686 (M2^) for notations.
Case (M28)
Figure 6.22
^ ^ ^ axe disjoint k-graphs.
Case (M29) , graph C^
Figure 6.25
C5 3 % *(a c)^B 2 ^ " * see Case (M2®) for notation.
Case (M 50) , graph . 40
Figure 6.24
^ 5 " " (P> Case (M3lJ , graph Eg . 1 * 1 * 2-'V- i r *—« Figure 6.25 V f% \ ' are disjoint k-graphs. C D-LW)2 5 Case (M 52) , graph Ej Figure 6.26 (^X y ^ ^ ° ^ a re 2 disjoint k-graphs. \ a p / \ 1 2 5 ' Case (M55J , graph . Figure 6.27 ( / y \ ) ' (l 2 3 ) 810 2 disJoint graphs. Case (M3*0 > graph Fg Figure 6.28 ( ^ a re 2 dis^oin'fc ^-graphs. \ x p 2 / \ a b c ' Case (M35) > graph G. y z. [3 c Figure 6.29 ( x y ? z ) 9 ( l 2 3) are disjoint k-graphs. 7 . SINKS ARE IRREDUCIBLE There axe among the 103 graphs, 38 sinks. To see that each sink K is nan-projective observe that either it is also a source, or there exist graphs such that ... ■+ K in the directed set l(P) (see section 8) w ith Y?" a source; hence since each source is non-projective by section 6, repetitive application of Lemma 2.1 proves that each sink is non-projective. We now show that each sink K is irreducible by proving that K - e C P for each e € e(k) , e(k) the set of all edges of K. For terminology, K non-projective, e £ e (k ) , e is called irreducible if K - e is projective. Observe that K irreducible if and only if e is irreducible for all e € e (K) . Different methods are employed to check irreducibility of these sinks. Accordingly, we group the 33 sinks into four classes. Class 1 consists of nine graphs, denoted by Case N1 to N9 , and is handled by Lemma 7*1 and Lemma 7*5 • Class 2 consists of eight graphs, denoted by Case N10 to N17 * We describe explicit embedding for K - e for each graph K and each edge e of K in class 2 • Class 3 consists of six cubic graphs denoted by Case N l8 to N23, irreducibility of which was proved by Glover and Huneke [ ^ ] • Class 4 consists of 15 graphs denoted by H-2 Case N 2h t o N 38. We use the irreducibility of graphs discussed in classes 1, 2, 3 and lemma 7*7 and Lemma 7*8 to assist In showing each graph in class 4 is irreducible* When describing an embedding of K - e explicitly the following lemma is useful* Lemma 7*1 Let K - (v^jVg) be embeded in P (hence P is divided into regions)* Let (R^, ..*, Rm) be regions containing V1 ' ^Sl* ***' Sn^ re 8iQns containing vg . If e is an edge common to the regions R. and S. for some i, j , then t) K - e C P . Proof: Using the embedding of K - (vi>v 2 ^ * Ri U U e is a single region for K - (v^Vg) - e CP* Since v^ Vg a re b o th in th e same reg io n R^ U U e , we can embed th e edge (v 1 ,Vg) in the region R^ U U e and this gives an embedding for K - e • We shall often represent a graph in P in such a way that there are exactly two edges crossing each other. These two edges w ill be called the crossing edges with respect to the representation* Clearly if (vi#v2^ is either °ae °f the crossing edges, then K - (v1 ,Vg) C P and by Lemma 7«1 each edge that is common to any two regions containing v^, Vg respectively is also irreducible* Lemma 7*2 Let K = L^ U Lg where either i) L^, Lg disjoint, or JfiJ. u ) ^ n Lg i s a vertex* If is projective and Lg is planar, then K is projective* For proof see Battle et al [ 1 ] . Lemma 7 .5 Let K = L^ U Lg where e ith e r i) L^, Lg disjoint or i i ) 1^ n Lg is a vertex* I f L ^L g € i ( r 2 ) , then K is irreducible. p Proof: L € l( R ) =* l is projective and L - e is planar* The conclusion follows immediately from Lemma 7*2 * Case N1 - N 5 The graphs Aj = V , Bx = Dig =» Kj ^111^ , Ej = Kjj v Kjj are irreducible by Lemma 7*3 ■ Lemma 7*4 Let K = L^ U Lg where L^ H Lg contains exactly two v e r tic e s , v ^V g • I f L^ U (v-^Vg) is projective and 1^ U (v-^Vg) is planar then K is projective* Proof: Since Lg U (v^/Vg) is planar it embeds into an open disc. An embedding of 1^ U (v^V g) in P extends to an embedding L^ U (v-^Vg) U Lg in P • Lemma 7.5 Let K = 1^ U Lg such that 1^ H 1^ = ^vi 'v 2 ^ where v^ is not valency 2 in L^ , i = 1, 2 and j = 1, 2 • Suppose i ) 1^ o r U ( v ^ V g ) € i ( r 2 ) ^ 5 i i ) L2 or L2 U (v^V g) € i ( r 2) iii) (v-^Vg) f K . Then K - e is projective for each edge e of K. Proof: Let e € e(k) , withait loss of generality suppose e € , Observe that L2 or Lg U (v^Vg) 6 i ( r 2) im p lies that Lg U (v^,Vg) is projective. Hence by Lemma it is sufficient to show that ( 1^ - e) U (v^,Vg) is planar. I f Iq _ U (v^Vg) € i( r2) , then Lj^ U (v-^Vg) - e Q is planar, hence the result. Hence, assume € l( 1R ) • / v If 1^=1^ then v^, v2 are vertices of (valency o f v^ n o t 2 ), thus (vi^v2^ CK a contradiction t o ( i i i ) . If = Kj ^ then v^, Vg cure non-adjacent vertices of Kj ^ , so (i^ U (v^/Vg)) - e has only five vertices (excluding vertex of degree 2 ) and has a trivalent vertex, so is hameomozphic to a proper subgraph of • Hence it is p la n a r. Cases N6 - N9 Lemma 7*5 proves the following graphs are irreducible. v F igu re 7*1 h6 V F ig u re 7*2 15 F ig u re 7*3 e6: F igure 7«^ We state the following lemma 'which w ill he used throughout the discussion of the following eight graphs* Lemma 7*6 Let K he represented in P with exactly two crossing edges then K - C E ^or eac^ crossing edges (v-j^Vg) . Suppose e is a common edge of regions R and S 1*7 where € r , v2 ^ S , then e is irreducible in K • Proof: It follows immediately from Lemma 7*1 • Case N10 V F igure 7«5 F igure 7*6 [(1 , 2 ) , ( r ,s ) ] SgJ C(p,q), (x,y), (b,c)] NOTE: (1) The six edges (p,s), (p,y), (q,r), (q,x), (r,l), (s,2) are irreducible (Figure 7*5 and Lemma 7*6). (2) The five edges (b,l), (b,2), (c,l), (c,x), (y,3) are irreducible (Figure 7*6 and Lemma 1*6)• (3) The two edges (a,2), (a,3) are irreducible (consider the representation form by crossing (b,l), (p,y) in Figure 7*6, then apply Lemma 7*6) • (*0 The three edges (p,r), (q,s), (a,l) are irreducible (by (1), (5), and . (3) The three edges (b,y), (c,2), (x,3) are irreducible (by (2) and Sg) • 48 Thus all 19 edges are irreducible in • So EL^ is irreducible* Case N i l *18: Figure 7-7 F igure 7*8 Sxs C(a,c), (x,y), (p*q)l Sg! C(2,z), (r,x,l>y)> (a>c,q,p)] Sy [(r,l), (x,y), (a,q), (c,p)] NOTE: (1) The five edges (b,3), (p,x), (p,3)> (q,r), (r,z) are irreducible (Lemma 7*8 apply to Figure 7»7)« (2) The three edges (p,r), (q,y), (o>3) are irreducible (apply S± on ( l ) ) . (3) The four edges (a,l), (c,y), (c,3), (z,l) are irreducible (apply on (l))* (4) The three edges (a,x), (a,3)> (c,l) are irreducible (apply S 1 on ( 3 ))« (5) The two edges (y>2), hence (x,2) by S1 , are irreducible (by Sg cm (z,l)) • 49 (6) The two edges (b,z), hence (b,2) by Sg , are irreducible (by Figure 7*8)* Thus all 19 edges in E^g are irreducible* Case M12 *16: F ig u re 7»10 S1: [(a,b), (2,3), (x,y)] SgJ [(l;2)j (q_>a)j (r>x)] S^s C(b, NOTE; (1) The four edges (p,x), (p,y), (q,r), (r,l) are irreducible (Figure 7*9 and Lemma 7*6)* (2) The three edges (p ,x ), (x,2) are irreducible (S2 on (1)) . (3) The two edges (b,y), (y>3) are irreducible (S^ on (l))* (4) The three edges (b,l), (c,2), ( (Lemma 7*6 an Figure 7*10)* (5 ) The three edges (a*3)> (b,2), (c,l) are irreducible (Sg on (4)). 50 (6) The three edges (a,l), (c,3), (q,2) axe irreducible (S^ on ( 4 )) . Thus all 18 edges are irreducible in E^g , hence E^g is irreducible* Case N13 B25: ' T t t f i SSV F igure 7 « H Figure 7*12 S1 : C (p ,l), ( S2: E(a,c), (y,z), (w,x), ( 2, 3 )] NOTE: (1) The four edges (b,3), (p,y), (a>x), (q,w) are irreducible (Lemma 7*6 cm Figure 7*H)* (2) The four edges (a,l), (b,2), (c,l), (p,z) are irreducible (S^, Sg on (l)). (3) The three edges (a,x), (p,l)> (x,y) are irreducible (Lemma 7*6 and Figure 7»12). (lj.) The four edges (a,3)> (w,c), (w,z), (z,3) are irreducible (by S.^ Sg an ( 3 ))* (5) The two edges (c,2), (y,2) are irreducible (by Sg on (4)). 51 (6) The two edges (h,l), (p,q.) are irreducible (for (p,q), embed thrcugh region denoted by @ in Figure 7*12 9 (b,l) follows by S^* So we have 19 edges are irreducible* Case N l4 DH ! Figure 7*13 Figure 7*1^ Figure 7»15 S: C(a,b), (p,c), (x,2)] NOTE: (1) The six edges (b,3), ( c ,z ), (c,2), (p,x), (p,z), (q,l) are irreducible (Lemma 1»6 on Figure 7«13)« (2) The eight edges (a,x), (a ,3), (b,2), (p,l)> (q,c), ( Figure 7*!1*)* (3) The three edges (a>l), (b,l), (z,3) are irreducible (Lemma 7*6 an Figure 7»15)» (4) The two edges (c,l), (p,q) are irreducible (by S on (2)). So all 19 edges are irreducible* 52 Case N 15. E2 8: Figure 7*16 C(p,q.)] S2: [(1,3)] Sj! C(a,b), (c,r), (x, 2 ) , ( p , 3 /Q jl)] NOTE: (1) The eight edges (a,q), (b,p), (b,q), (b,3), (c,3), (p,r), (x,r), (x,2) are irreducible by Figure 7-16 • (2) The four edges (a,p), (b,l), (c,l), (q,r) are irreducible (by S^, Sg on (l))« (3) The three edges (a,l), (a,3), (c,2) are Irreducible (by S^ on ( l ) ) . (4) The two edges (a,x), (b,2) are irreducible (represent Egg by crossing (p,b) with (a,q) from Figure 7*16 and apply Lemma 7*6 to (p,b) , we get that (a,x) is irreducible, (b, 2 ) follows by applying S^ on (a,x))« Thus all 17 edges are irreducible. 53 Case N l6 g ur 71Fgure 7*17 F ig u re 7*18Fig Figure 7*19 Figure 7»20 S a [(q>b), (p> 3 )> ( a ,r ) ] K: (1) (b,w), (b,3), (q,z), (w,l) are irreducible (Figure 7»17)» (2) (b,z), (p,q), (q,w) are irreducible (by S on (l))* (3) (a,y), (p>r), (x,r), (x,l) are irreducible (Figure 7«l8)« (*0 (a,x), (a,3), (r,y) are irreducible (by S on (3)). (5) (c,p), (y,2), (z,2) are irreducible (Figure 7«19)» (6 ) (c,3) is irreducible (S on (5))« (7 ) (c,l) is irreducible (Figure 7«20)« (8 ) ( c ,2 ) is irreducible (by placing a in region ® in Figure 7*18, cross (a,3) with (c.,2)). Thus all 20 edges are Irreducible* Case N17 2 0 * Figure 7*21 Figure 7*22 Sj^s [(a,d), (b,c), (2,4), (3*5)* ( 1 * 6)] S2: C(2,3)] S y [(^ ,5 )1 NOTE: (1) (c , 6), (d,4), (b,4), (a, 6 ) are irreducible by Figure 7-22 and Lemma 7»6 • (2 ) (d ,5 )* (b , 5 ) are irreducible (by and (l))* (3 ) (d* 6) is irreducible (by placing 6 on ) . (4) (c,4) is irreducible (by placing 4 on (^ ) • (5) (c*5) Is irreducible (by and (4))* (6) All other edges are irreducible by and (1), (2), (3), (4), (5) . Let L, K be graphs such that L =* • Then there is a natural correspondence between edges and vertices of L and K • 55 In the discussion that follows, no distinction will be made unless it is necessary for clarity* We have been calling a vertex cubic if it is of valency 3 • Recall an edge e is called cubic of (v^/v^v^) ° is in a 3 cycle (v-,v_,v_) and if e » (v_ ,v_) say* and v, is X d. 3 X <£ j cubic . Lemma 7.7 Let K be nanTprojective, L = ^(K) ” e where e is cubic of (v 1 *Vg,v^) in ^(K) . If L is irreducible, then all edges of K, except possibly the 3 edges on (v^,v 2 ,v ^ ) a re irreducilbe* Proof; Let e' not be in the 3-cycle • Suppose K - e ' cf P • Then - e ' ^ P by Lemma 1.1. Hence - e' - e 9^ P by Lemma 3«^ * as e is cubic. But then L - e* = (^K - e) - e 1 9^ P which contradicts that L is irreducible, so e' is irreducible on K. Lemma 7.8 Let K be nan-projective. L = ^(X) “ (e^,e') where e is cubic of (v 1 ,Vg,v^) in <^K aad e' is cubic of (v^,Vg,v^) in «^K-e . If L is irreducible then if e is nob an edge in the two 3-cycles (v-^Vg/v^), (v^,Vg,v£). then e is irreducible. Proof: Suppose e is an edge of K not in (v^,v or (v^,Vg,Vj) such that K - e S^K ■ e - a f P by Lemma 3«^> a s e i s cubic in • Hence «^K - e - e - e' rf P by Lemma as e is cubic in §yK - e . Hence L =* - (e*e'} - ' e r f P , which contradicts the irreducibility of L • We shall now apply Lemma 7*7 and Lemma 7*3 to check irreducibility of the following graphs* In all cases either 3 o r 6 edges remain to check* For completeness* we first lis t the following trivalent graphs for our reference* Cases Nl8-N2^ The graphs F^, F^, F ^ , F ^ , F^, G are irreducible ( see Glover and Huneke [ 4 ]) . Case Figure 7»23 S: C(x,y), (1*2)] Since Sxs(p a)(^) " (P>*) s Fio ' since P10 F12 ^see se c tio n 8 ) and F^2 is irreducible (see Case N19) * F 10 i s irreducible* Hence by Lemma 7*7 we only need to check the three edges 57 (a,x) ~ (a,y) "by S and (a,y) f & is irreducible, hence (a,x) is irreducible* Similarly (p,x) ~ (p,y) by S is irreducible* Case N25 d1 9 : Figure 7*24 S - C(a,l), (b,2), (c,3), (d,4)] Note th a t \ . ( 2 3 *7 ^ u t *7 * *9 ( see se c tio n 8 ). Now irreducible E^ irreducible. Hence by Lemma 7*7, we need only check the three edges {(1,2), (1,4), (2,4)} • But by symmetry S we have (1 , 2 ) - (a ,b ) (1.4) ~(a,d) (2.4) ~(b,d) so all edges are irreducible* 58 Case N 2.6 \ r Figure 7«25 C(a,x), (p,z), (1,5)1 S2: C(a,b), (y,l), (z, 2 )] note that ®.(p, 2 ) ( Di 7 > ' (p,2) = E26 and E26 *' *25 (see section 8) and 3^ irreducible (Case N13) ^ ^ 5 irreducible* Now applying Lemma 7*7, we only have to check three edges {(z,p), (2,z), (p,2)} « Now (p,2) is clearly irreducible by Figure 7-25 and (p,z) ~ (p,2) (by Sg) and (z,2) ~ (p,2) (by S1). Therefore all edges are irreducible* Case N 27 * 1 6 ' Figure 7*26 S: C(p,q), (b ,c )] 59 Note that Sa; (p^ " (P> E^q •*■ (see section 8). Since EL^ is irreducible (Case N10) , we have E^q is irreducible. But then by Lemma 7*7 we need only to check irreducibility of edges {(a,p), (a,q), (p,a)) but clearly (a,q) is irreducible (Figure 7*26) and (a,p) ~ (a,q) (by s) is also irreducible. To see (p,q) is irreducible, we place p in region (§) in Figure 7*26 and crossing only (p,q) and (x,2) • Case N28 Figure 7*27 S: [( p ,q )] Note that \>.(p ~ (P*0 53 % * % is irreducible by Case N9 • So by Lemma 7*7 only (p,y), (q>y)> (p>q) need to be checked. But (p ,y ) is irreducible by Figure 7*27 • (q,y) = (p,y) because p ~q. For (p,q) > we place q in the region © in Figure 7*27 and crossing only (p,q) and (y,z) • 60 Case N 29 Figure 7*28 S: C(q,b), (x,2), (z,l)3 Note that ^.(p^O fo) " CCp^Oj- (2,b)} = F^ . Since F ^ is irreducible by Lemma 7*8 we need only check irreducibility of C(b,2), (b,p), (p,q), (p,x), (q,x), (p,2)} . Now (q,x), (b*p)j (p,2) are irreducible by Lemma 7*6 > F igure 7 * 28, but then (b,2), (p,q), (p>x) are irreducible by S • Case N50 Figure 7*29 S: t(p,q), (b,c), (x,y)] Note that sa.(p,q)(^) “ ^ 2 8,14 *32 * *33 ^ ^ ^see section 8). Now E,i, irreducible (Case Nl6) IL0 irreducible* 6 l Now applying Lemma 7*7, we only need to check the following edges ((a,p), (a,q )> (P>q)} • But (a,q) is irreducible (froom Figure 7*29) and (a,p) ~ (a,q) , hence irreducible* To see (p,q) is irreducible, place p in region (§) in Figure 7*29 and cross only (a,2) and (p,q) • Case N31 Figure 7*30 S: C(p,q)l Note t h a t sx : (p#<1)(D2 ) “ (p,q) “ ^ 8 8116 *28 i s irreducible (by Case N15) • So applying Lemma 7*7, we need only check the three edges C(p,q), (p,x), (q,x)} , but (p,x) is irreducible (from Figure 7*30) and (q,x) ~ (p,x) hence is irreducible* To see (p,q) is irreducible, place p in the region (g) crossing only (p,q) and (x,2) . 62 ra s e N32 Figure 7*31 S: C(3,l)> 0>A), (2,a), (4,c)] Note that Si.(2,4)(Cl()) “ = Dg • Dg is irreducible hy Case N 31 • Applying Lemma 7„7 , we need only check three edges {(1,2), (1,4), (2,4)}. But hy symmetry S we have (1.4) ~ (b,c) (2.4) ~ (a,c) (1,2) ~ (a,b) hence all are irreducible. Case N 53 Figure 7 .3 2 S: [(4,6), (1,3)1 63 Note that \ . ( X^)(C9) " U ^) » Dg and Dg ■*■ P.,■) (see section 8). 1 irreducihle (Case N14) =* Dg irreducible. Now applying Lemma 7*7, we have only to check {(1**0* C1-**), (4,x)} , b u t by S (1A) - (3,6) (l,x) ~ (3,x) (ij-,x) '■*' (6,x) So a l l edges a re irre d u c ib le * Case N ^ Figure 7*33 S: C(l,2), (x,y)] Note that sx.(p i )( ° q ) “ (p A) « is irreducible (Case N26) • So apply Lemma 7*7, we need only check the edges {(x,p), (x,l), (p,l)) * Now by S (x,p) ~ (y,p) (x,l) ~ (y, 2 ) (p,l) ~ (P,2) So all are irreducible* Figure 7*34 Figure 7*35 Sjl; [(4,5), (2,3)3 S2: [(5,6), (1,2)3 Note that %.( 4 ,5 ) ( Bi 2 ) " (^5) » is irreducible (Case N33) . So by Lemma 7»7, we only have to check three edges ((4,5), (6,5), (6,4)} . Now sx, S2 =» (4,5) ~ (4,6) ~ (5 , 6 ) , but Figure 7»35 shows (4,5) is irreducible* Case N 36 Figure 7*36 Si C(C,E), (A',y), (A,z), (B,D), (t,x)3 Note that '^.•(c,D)^ll^ ” C(C,D), (x,y)} — ^ . q • ®].o irreducible (Case N29) • Now by Lemma 7*8 we only have to check 65 s ix edges & =. {(A,C), (A,d), (D,d), (D,x), (D,y), (x,y)} but by S (A,C) (E#z) (A,D) ~ ( B , z ) (Cj D) ~ ( b , e ) (D,x) ~(B,t) (D,y) ~(A',B) (xjy) ~(A ',t) So all edges are irreducible* Case N 37 9* Figure 7*57 Figure 7*58 S1: C(p,A), (D,C), (AS e )] Sg: t(x,y)] Note that s;A.(x,b)0b9) - ((B,x), (E,y)} = J>9 • J>9 i s irreducible (by Case N30) • So by Lemma 7*8 we need only check six edges = ((A',x), (A',B), (B,x ) , (B ,e ), (B,y), (y,E)} • But (A',x) = (A',y) is irreducible by Sg , (y>E) ^ (x,E) is irreducible* (A' j B)* (B,y) are irreducible by placing B in (g) 66 on F igure 7 .3 8 and (B,E) - (A*,B) , (B,x) — (B,y) by S1 , so are irreducible* Case N 38 Figure 7*39 Sx: C(l,6)] S2: [(2,5)3 Note that ( 2 , 3 ) 0 ^ “ (2 ,3 ) 53 B12 which is irreducible by Case N 35 . So applying Lemma 7*7, we check only three edges {(1 , 2 ) , (1 , 3 ) , (2 , 3 )} , but by S1, S 2 (1,2) -(2,6) (1.3) -(3,6) (2.3) -(3,5) Hence evexy edge is irreducible* 8 . SOURCES AND SINKS OF A GRAPH By the definitions of a source and a sink, any graph K in the directed set l(p) is either a source or follows a source with the same Betti number as K, and also is either a sink or is followed by a sink with the same Betti number as K • Sources and sinks related to K in this fashion are the sources and sinks of the graph K • This section includes some branches of l(P ) demonstrating some sources and sinks for each graph in l(p) which is not both a source and a sink. More specifically we will use the notation * To!(y - — ....— to indicate the following information: (1) X?" is a source of K* j (2) f isa sink of K1 ; (5 ) K? = § , vK1 and similarly for K?" , K* ^ J v0 a v * ** * t) (W) The notation for vertices vQ, agree with labelling of K 1 as appears in the case discussing K? in section 6 ; and (5 ) I f VqiCv^, vA) is replaced by P th en P i s assumed to be the outpoint of and the splitting 68 is left to the reader to compare. p -> O- -9 a : (1,2,4) 5 : (1,2,4) B, Ba B,, : (a,p>a) 5 =V c.'1 c. ’3 Figure 8 .1 l:(a,b,c) 2:(a,b,e) ©------> .... o ------^ ------© % c6 <7 l:(p,b) a o 2:(b,c) a:(l,3) >------o ------> ------© D. 12 D15 1 4 Figure 8 .2 70 P P ^ - ■■■— ■ ^ ■1 ■ *L % *5 2 :( b ,c ) a s ( l,5 ) > - -O- — >------© E, *9 *8 c:(p,5*) 2:(q,a') '22 '20 '21 Figure 8.3 7 1 X :(a ,c ) 2!(< ^ c ) l : ( p ,q ) 10 c :( 2 ,x ) 2 : ( r ,a ) Figure 8 .4 72 c : ( q ,l ) 12 10 11 Figure 8*5 9* CONJECTURES AND RELATED QUESTIONS We have shown that l(P) contains 103 graphs* However, we conjecture that l(p) Is the set consisting of exactly the 103 graphs of the Appendix* The proof of this conjecture should be made in the following way* In [ 3 3 Huneke and Glover w ill show that A^, A^, Bg, B^, Dg 0 are the maximal elements in l(P) • In [11 ] the writer will show t h a t th e 103 graphs above are all graphs below these five* A first question that can be asked about the partially ordered set l(P) is whether it is essentially a lattice* That is, after introducing a maximal and minimal element in l(p) do we have a lattice* The answer to this question is no. The following partially order subset of l(p) shows this* E. '15 F igure 9«1 More particularly E^ and E^ do not have a supremum in l(P) • Another natural question one may ask about l(M) is whether for K € l(M) there exists an immersion (a local embedding) f : K -*■ M with a single double point interior to two edges* This 75 appears to be correct for M = P and K any one of the 103 graphs of our list. There is no proof of this observation about l(P), or general I (M) . It is conceivable that |l(M )| increases rapidly as a function of the genus of M . In particular, one may ask: Is l(M) finite for all M ? The question is not yet answered. In view of the last paragraph, a more practical way of attacking the problem of characterizing l(M) would be to look for an effective algorithm for generating graphs in l(M) . There has been very 2 little success with the exception of l(lR ) . Rather than enumerating all elements in l(M) it seems to be more practical to look for a ll maximal elements in l(M) or to derive a characterization for all maximal elements in l(M). APPENDIX: THE 103 GRAPHS IN l(P ) The following 103 graphs are given by the order , (1 < i < h ); B± , (1 < i < 12); C.. , (1 < i < 10); D± , (1 < i < 20); E± , (1 < i < «.); F± , (1 < i < 15); and G . A^*s are graphs with Betti number £ =• 12 • Similarly B^'a, Ci*s, Di 's, *s, F^s, G are graphs with fj = 11, 10, 9, 8, 7, 6 respectively. The valency sequence of each graph is given directly below the picture. The general form of the valency sequence is Sq Cs^ Sg, ..., sn) where sQ =* n is the number of vertices of the graph and s^'s are valency of vertices arranged in non increasing order. 75 76 A1 : 9 ( 8 , b, k, b, b, b, b, b, i t ) 1 0 ( 6, i f , k, b, b, i t , b, it, it, if) l l ( bf i t , b, i t , b, b, b, i t , b, it- , it-) 77 7 (6, 5, 5, 5, 5, 5, 5) 1 0 ( 14-, k, *4, b, h, b, k, b, k, i4) * 9(6, b, b, b, b, b, b, b, !<-) 10(1*, 1*, b, b, b, b, b, b, 1+, M (£ ‘H ^ ^ ^ *9 ‘9)9 in ‘n ‘n ‘n ‘9 ‘9)9 in ‘n ‘n ^ €m ‘*i ‘9 ‘9)9 80 9(6, b, b, %, b, k, K, b, b) 10(1^, U, b, b, b, b, b, b, b, b) 81 8(6, 5, 5, 5, b, b, k, 3) 10(7, k, b, b, b, 3, 3, 3, 3, 3) 11(5, b, b, A, if, b, 3, 3, 3, 3, 3) 82 11(6, 4, 4, 4, 4, 3, 3, 3, 3, 3t 3) 12(4, 4, 4, 4, 4, 4, 3, 3, 3# 3, 3 , 3) 9(6, 6, 4, 4, 4, 3, 3# 3, 3) 83 10(6, k, k, k, k, h, 3, 3, 3, 3) ll(it-, 4, k, h, k, k, it-, 3, 3, 3, 3) 9(6, 5, ^ ^ k> 3, 3, 3) 9(6, k, K, k, k, k, k, 3, 3) 8(5, 5, 1+) 5, 5, ^ K 3, 3, 3) 8 5 9(5, 5, 3, 3, 3, 3) 3 86 9(5, 5, ^ 3, 3, 3, 3) 10(5, 3, 3, 3, 3, 3) 87 k 3 88 Dn : 1 1 ( 5 , K 3, 3, 3, 3, 3, 3, 3) D12 : D1 3 : 10(5, h, k, k, 3, 3, 3, 3, 3) 89 DlU : 3 D 1 5 : 10(6, 6, 3, 3, 3, 3, 3, 3, 3, 3) Dl6 : 9(5> ^ k> 3> 3> 3^ 9 0 D,„ : 17 10(6, 3, 3, 3, 3, 3, 3) d18 : 1 1 ( 1*, k > 3' 3' 3' 3 * 3' 3) D1 9 : 9 1 10(5* 5* ^ 3, 3* 3, 3, 3, 3) 92 I • ’X * 11(6, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3) 12(5, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3) u ------1 Eg : ------g) \ I t *— ------—g> 4r------i > / \ < 13(^, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3) 9 3 10(5, 5, 3, 3, 3, 3, 3, 3, 3, 3) 11(5, 4, 3, 3, 3, 3, 3, 3, 3, 3, 3) 12(4, 4, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3) 9 (^ k> k> k> 3, 3, 3, 3) 9 (J+, k, k, k, k, 3, 3, 3, 3) 95 8(5, 5, 5, 3, 3, 3, 3, 3) 9(5, 5, 3, 3, 3, 3, 3, 3) 3, 3) 9 6 E13 : 10(5, 3, 3, 3, 3, 3, 3, 3) E l l ; : 3 E15 : 9 7 E16 : 3 E17 : El8 : 98 19 • 12(^, 3, 3, 3, 3f 3, 3, 3, 3f 3, E20 : Q(kf k, k, k, k, 3, 3) 9(*N ^ ^ 99 E22 ! 10(1^, K, b, b, 3, 3, 3, 3, 3, 3) E23 5 3 E2U : 3, 3, 3, 3, 3) 100 25 12(^f, 3j 3, 3, 3, 3) 3j 3t 3> 3, 3) E2 6 : 11(6, 3, 3, 3, 3> 3, 3, 3, 3, 3> 3) E, 27 9(5, 5, k, 3, 3, 3, 3, 3, 3) 101 E28 : 10(5, 5, 3, 3, 3, 3, 3, 3, 3, 3) E 29 10(5, 3, 3, 3, 3, 3, 3, 3) E3 0 : 102 ^ ^ 3, 3, 3, 3, 3, 3, 3, 3) 11(5, b, 3, 3, 3, 3, 3, 3, 3, 3, 3) ■o- 12(^, h, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3) 1 0 3 3 9(^j 3/ 3, 3) E 3 6 • 3 k 3, 3, 3, 3, 3, 3) K k 105 •O- h o : 1 0 ( ^ b, b, b, 3, 3, 3, 3, 3, 3) H ( ^ b, 3, 3, 3, 3, 3, 3, 3, 3) F1 ; 106 10(4, 4, 3, 3, 3, 3, 3, 3, 3, 3) 10(4, 4, 3, 3, 3, 3, 3, 3, 3, 3) 10(4, 4, 3, 3, 3, 3, 3, 3, 3, 3) 3, 3 ■6- 10(^, k, 3, 3, 3, 3, 3, 3, 3, 3) i 108 11(1*, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3) -6- lit1*, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3) llC 1*, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3) 109 12(3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3) 12(3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3) f 110 12(3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3) <) 3, 3, 3, 3, 3, 3, 3, 3, 3) BIBLIOGRAPHY 1 . B a ttle , J . , H arary, P ., Kodama, Y ., Youngs, Y. W . T., Additivity of the genus of a graph, BAMS 68 (1962), pp. 565 -5 6 8 . 2. Glover, H. and Huneke, J ., A Haefliger theorem for embedding graphs in the projective plane. (To appear) 5 . ______, A characterization of graphs that embed in the projective plane, (in preparation) 4. , Cubic irreducible graphs for the projective plane.(To app ear) 5. Heawood, P. J., Map color theorems, Quart. J. Math. 24 (I 89O), PP. 332-358. 6. Kainen, P., Embedding and orientation of graphs, combinatorial structures and their applications, Proc. Calgary International Conference, Calgary, A lta., ( 1969), PP* 193-196. 7 . Kuratowski, K., Surle problems des courbes gauches en tcpologie, Func. Math. 15 (1930 )> PP« 271- 283. 8. Ring e l, G ., Map Color Theorem. Die Grundehren der mathematischen, W issens chaften in Einzeldarstellungen, Band 209, Soringer- Verlag New York H eidelberg B erlin, 1974. 9* ______, Farbensatz fur nichtorientierbare Flachen beliebigen Geschlechts. J . re in e Angew Math. 190, pp. 129-147 (1952). 10. Ringel, G. and Youngs, J. W. T., Solution of the Heawood Map - Coloring problem, Proc. Nat. Acad. Sci. USA 60 ( 1968) , pp. 438-445. 11. W a n g , C. S ., A Kuratowski theorem for the projective plane, (in preparation) 111 INDEX OF GRAPHS IN l(P ) A “ G r a p h s C - G r a p h s \ , 3$ , 68, 7 3, 76 / 36, 68, 81 Aa , 6 8, 76 Cg , 68, 8l A^ , 44, 6 8, 76 , 68, 82 A4 , 33/ 66, 73/ 77 Cjj_ , 44, 68, 82 , 39/ 40, 6 9, 82 B - G r a p h s 18, B1 ' 36, 44, 77 c6 , 69, 83 Bg , 17/ 39, 68, 73, 77 G f / 46, 69, 83 1 8, 6 8, 78 Cq , 31*, 63, 83 B3 * / 34, 62, 64, 84 b4 ' 1 8, 1*5, 68, 78 32, 6 8, GjQt 3 6, 62, 84 B5 ' 73, 78 17, 6 8, 79 b6 ' D-Graphs By / 17, 6 8, 79 / 31/ 35/ 69, 84 b8 ' 6 8, 79 D2 , 19, 6 1, 6 2, 6 9, 85 B^ / 65, 68 , 80 / 19/ 6 9, 85 B10' 18 , 68 , 80 , 19/ 6 9, 85 BH> 18 , 64, 68 , 80 / 19, 69/ 86 B12' 32, 64, 34, 66, 81 D6 , 19, 69, 86 , 2 0 , 6 9, 86 Ifc / 19/ 63, 6 9, 87 112 113 D-Graphs (continued) E“Graphs (continued) 2 0 , 6o, v 65, 69, 87 ®LL' 2 1 , 71, 95 2 0 , 6 0, D1 0 ' 69, 87 * 1 2 ' 2 1 , 71, 95 63, 6 9, 88 2 2 , 71, 96 Dii ' 51, V 8 8 2 2 , 96 D1 2 ' 19, 38, 69, *14' 71, 88 24, Biy 19, 69, V 59, 71, 73, 96 2 0 , 24, 7 0 , 97 Dl4 ' 59, 69, 89 =16' 49, 46, 71, Biy 39, 89 V 97 2 2 , 48, Dl 6 ' 37, 58, 89 =L8 ' 71, 97 34, 58, 35, 6 3, 9° 24, 47, 59, 71, 98 d ^ , V 2 0 , 44, 90 70, 98 ^ 8 ' 36, E2 0 ' 30, 2 2 , 70, 98 &L9' 37, 57, 90 =2 1* 2 0 , D2 0 ' 37, 54, 73, 91 E2 2 ' 23, 70, 99 23, 70, 99 E-Graphs E23' 24, 70, 99 Bl , 2 1 , 36, 70, 92 E24' 7 0 , 100 2 2 , e25, 24, 50, %' 70, 92 5 8 , 7 0 , 1 0 0 44, E26 ’ 2 1 , 35, h >25, 70, 92 2 1 , 35, 7 1 , 100 Eif *. 2 1 , 38, 70, 93 V 2 1 , 52, 6 1, 7 1 , 101 70, 93 E2 8' b ' 24, 2 2 , 71, 101 e6 ' 46, 59, 70, 93 E29' 24, 101 2 2 , 40, 70, 94 71, *7 ' 57, V 24, 71, 102 Eg , 2 2 , 40, 70, 94 V 6 0, 102 23, 56, 57, 70, 94 V 71, b ' 24, 6 0, 7 1, 102 \o ' 29, 71, 95 b y 25, 53, 6 0, 71, 103 m E-Graphs (continued) G-Graph '&y=>t 2 2 , 31, 71/ 103 Ejq , 23, 71, 104 E^9, 2 2 , 3 8, 71, 73, 104 E4 0 ' 25> 38' 59, 71, 73, 105 E41, 24, 71, 73, 1P5 F-Graphs F1 , 32, 72, 105 F2 , 25, 72, 106 * 3 , 25, 72, 106 * 4 , 25, 72, 106 *5 , 25, 40, 72, 107 F6 , 25, 41, 72, 107 F? , 26, 72, 107 Fg , 26, 72, 108 F9 , 2 6, 72, 108 * 1 0 ' 2 6, 5 6 , 7 2, 108 F1X, 2 6, 72, 109 * 1 2 ' 26, 5 6 , 72, 109 f13, 2 6, 5 6 , 7 2 , 109 ' 1* ' 2 6, 5 6 , 6 0, 7 2, n o * 1 5 ’ 2 6, 3 6, 5 6 , 1 1 0