On the Connectivity of Graphs Embedded in Surfaces
Journal of Combinatorial Theory, Series B TB1809 Journal of Combinatorial Theory, Series B 72, 208228 (1998) Article No. TB971809
On the Connectivity of Graphs Embedded in Surfaces Michael D. Plummer* Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37240
and Xiaoya Zha Department of Mathematical Sciences, Middle Tennessee State University, Murfreesboro, Tennessee 37132 Received July 16, 1996
In a 1973 paper, Cooke obtained an upper bound on the possible connectivity of a graph embedded in a surface (orientable or nonorientable) of fixed genus. Furthermore, he claimed that for each orientable genus #>0 (respectively, nonorientable genus # >0, # {2) there is a complete graph of orientable genus # (respectively, nonorientable genus # ) and having connectivity attaining his bound. It is false that there is a complete graph of genus # (respectively, nonorientable genus # ), for every # (respectively # ) and that is the starting point of the present paper. Ringel and Youngs did show that for each #>0 (respectively, # >0, # {2) there is a complete graph K n which embeds in S # (respectively N # ) such that n is the chromatic number of surface S # (respectively, the chromatic number of surface N # ). One then easily observes that the connectivity of this K n attains the upper bound found by Cook. This leads us to define two kinds of connectivity bound for each orientable (or nonorientable) surface. We define the maximum connectivity } max of the orientable surface S # to be the maximum connectivity of any graph embeddable in the surface and the genus connectivity } gen(S # ) of the surface to be the maximum connectivity of any graph which genus embeds in the surface. For nonorientable surfaces, the bounds } max(N # ) and } gen(N # ) are defined similarly. In this paper we first study the uniqueness of graphs possessing connectivity } max(S # ) or } max(N # ). The remainder of the paper is devoted to the study of the spectrum of values of genera in the intervals [#(K n )+1, #(K n+1 )] and [#(K n )+1, #(K n+1 )] with respect to their genus and maximum connectivities. 1998 Academic Press
* Work supported by ONR Contract N00014-91-J-1142. Corresponding author. Work supported by NSF Grant DMS-9622780. Work supported by NSF Grant DMS-9622780.
208 0095-895698 25.00 Copyright 1998 by Academic Press All rights of reproduction in any form reserved.
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1. INTRODUCTION Throughout this paper, }(G) will denote the (vertex-)connectivity of the graph G. Similarly, denote by #(G) (respectively, #(G)) the orientable (resp. nonorientable) genus of G. For any real number x, denote by WxX the least integer greater than or equal to x and by wxx the greatest integer less than or equal to x. The maximum connectivity of a surface (orientable or nonorientable) is defined to be the maximum value of connectivity taken over all graphs embeddable in the surface. We denote this parameter by } max(S # ) when S # is orientable and by } max(N # ) when N # is nonorientable. It is well known that } max =5 for both the plane and the projective plane. In his 1973 paper, Cook [2] proved the following two results. Theorem 1.1. (a) if G is any nonplanar graph and if G is embedded in the orientable surface S # , then }(G)
\
5+- 1+48# , 2
while (b) if G has nonorientable genus at least 2 and if G is embedded in the nonorientable surface N # , then }(G)
\
5+- 1+24# . 2
Note that the bound in part (b) of the above theorem also holds for graphs having nonorientable genus 1. Cook then claimed that, by results of Ringel and Youngs [14] (resp. Ringel [12]; see also [13]), for each value of #>0 (resp., # >0, # {2), there is a complete graph K p embeddable in S # (resp. N # ) such that }(K p )=} max(S # ) (resp. }(K p )=} max(N # )). Therefore, } max(S # )=w(5+- 1+48#)2x and } max(N # )=w(5+- 1+24# )2x. Our first question is: Are these complete graphs the only graphs which attain Cook's bounds for } max(S # ) (resp. for } max(N # ))? It is well known that there are infinitely many 5-connected graphs which are planar and infinitely many 5-connected graphs with nonorientable genus 1. It has also been shown by Negami [810] that there are infinitely many 6-connected graphs embeddable in each closed surface with nonpositive Euler characteristic. On the other hand, it is also well known (see, for example, [16]) that any 7-connected graph embeddable in a surface of
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genus # or # has size bounded above by a function of the genus and hence there are at most only a finite number of 7-connected graphs embeddable in any surface, orientable or nonorientable. Thus for any surface with # 2 (resp. with # 3), the number of graphs embeddable in that surface and having connectivity }, 7 } } max , is finite. Furthermore, Cook also claimed that the complete graphs attaining his bounds for surfaces S # and N # are genus embeddable there. This is not necessarily the case. It is well known, in fact, that as genus increases, there are many surfaces which do not admit a genus embedding of any complete graph. The maximum connectivity of any graph having orientable (resp. nonorientable) genus # (resp. # ) will be denoted by } gen(S # ) (resp. } gen(N # )) and will be called the genus connectivity of the surface S # (resp. of surface N # ). Clearly, for any orientable surface S # (resp. nonorientable surface N # ) } gen(S # ) } max(S # ) and } gen(N # ) } max(N # ). But is there a surface with genus connectivity strictly less than its maximum connectivity? And precisely when does strict inequality hold in these inequalities? It is obvious that } max(S # ) and } max(N # ) are monotone nondecreasing functions of # and #, respectively. However, it will be shown that, surprisingly, when the genus increases, } gen may in fact decrease. For example, it will be shown later that } gen(S 35 )=23, but } gen(S 36 ) & = . 12 12 2 12 But then solving this inequality for r, we obtain r(n&6) 11 and since n&6>0, it follows that r 11(n&6). But then for n 18, r1 such that there exists a Class B surface in the interval [ g+1, g+x] (resp. in the interval [ g +1, g +x]).
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Let us begin our study of the breaking point by providing a lower bound. For all integers m 12, write m=12s+k, where s 1 and 0 k 11. Consider first the orientable case. Define # s, k =#(K m )=#(K 12s+k ). By the RingelYoungs genus formula [13, 14], # s, k =12s 2 +(2k&7) s+ W(k&3)(k&4)12X. Next, define $ k and = k as $k =
if k=1, 2, or 5 otherwise
&2,
{ &1,
and = k by
{
&1,
if
= k = 0, 1,
k=2,
if k=0, 1, 3, 5, or 6, otherwise.
Note that # s, k+1 &# s, k =2s+= k . To see this, note that the difference # s, k+1 &# s, k =12s 2 +(2k&5) s+W(k&2)(k&3)12X&(12s 2 +(2k&7) s+ W(k&3)(k&4)12)X)=2s+W(k&2)(k&3)12X W[(k&3)(k&4)12X. Upon computing the difference of the two ceiling functions, observe that # s, k+1 & # s, k =2s+= k , as claimed. We then have the following theorem. Theorem 3.1. Suppose that m=12s+k and that K m genus embeds in surface S #s, k so that } max(S #s, k )=} gen(S #s, k )=m&1. Suppose $ k is as defined above and suppose also that s+$ k 1. Then if i is an integer in the interval I s, k =[# s, k +1, # s, k +s+$ k ], S i is in Class B (and hence B{# s, k . It is also easy to check that for these same values, #(O(12s+k+1))=W f s, k(1)X . On the other hand, just as in the proof of Theorem 3.2, it may be shown that W f s, k(1)X g A . Finally, since #(O(m+1))># s, k , then by the definition of g A , g A #(O(m+1)). So assembling these inequalities, W f s, k(1)X g A #(O(12s+k+1))= W f s, k(1)X and the equality must therefore hold. K Consider now a possible extension of the preceding theorem to the case when s=0. Since #(K 1 )= } } } =#(K 4 )=0 and #(K 5 )=#(K 6 )=1, g A is not defined for k=1, 3, and 5. Suppose k=7. Then #(K 7 )=3 and #(K 5 )=4, so g A =4. But #(0(8))=3, so equality does not hold. Similarly, when k=11, #(K 11 )=10, #(K 12 )=12, and g A 11, but #(O(12))=10, so again equality does not hold. However, when k=9, #(K 9 )=5, #(K 10 )=7, and #(O(10))=6. Thus in this case #(O(10))= g A . Note that there are four of the 12 possible values of k which are not covered by Theorem 5.1 and there are six of the 12 possible values of k not covered by Theorem 5.2. Conjecture. In each of the 10 unsettled cases above, the breaking point is the orientable (nonorientable) genus of O(12s+k+1). In Section 2 we considered those surfaces in which the complete graphs genus embed and considered the problem of deciding when these complete graphs are the only graphs embeddable in these surfaces having connectivity } max for that surface. Of course, if we drop the demand that the complete graphs genus embed in a surface, we can look at the more general question as to when for any surface is a complete graph the only graph which embeds there and has connectivity } max . We now conclude by addressing this more general question. We begin with the orientable case.
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Recall that #(K 12s+k+1 )&#(K 12s+k )=2s+= k . Consider the surfaces S#s, k +r for 0 r 2s+= k &1. For all these surfaces, } max(S #s, k +r )=m&1= 12s+k&1. In the next theorem, it will be shown that for four of the 12 congruence classes modulo 12, one can completely decide when the graphs having connectivity } max are the only graphs embeddable in the corresponding surface. Theorem 5.3. If the surfaces S #s, k +r and the quantities $ k are defined as above, and if s+$ k 1, then (a) if 0 r s+$ k , K m is the unique graph with connectivity m&1 which embeds in S #s, k +r , while (b) if s+$ k +1 r 2s+= k &1, and k=1, 5, 7, or 11, then there is more than one graph with connectivity m&1 which embeds in S #s, k +r Proof. Suppose G{K m is a second graph which embeds in S #s, k +r and G has connectivity m&1. Then G must have at least m+1 vertices and so by inequality (3.1), #(G) f s, k(1)=#s, k +s+$ k +1 #s, k +r, a contradiction, and (a) is proved. To prove part (b), observe that, of course, Km is one graph which embeds in S#s, k and, hence, also in S#s, k +r for all r 0. By Theorem 2.2, #(O(12s+k+1)) =W((6s+(k&1)2)(6s+(k&5)2)3X and for k=1, 5, 7, and 11 and for these four values, the reader can easily show that this quantity, in turn, equals # s, k +s+$ k +1. So, for r=s+$ k +1 there is a second graph with }=m&1. K If s+$ k
On the Connectivity of Graphs Embedded in Surfaces Michael D. Plummer* Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37240
and Xiaoya Zha Department of Mathematical Sciences, Middle Tennessee State University, Murfreesboro, Tennessee 37132 Received July 16, 1996
In a 1973 paper, Cooke obtained an upper bound on the possible connectivity of a graph embedded in a surface (orientable or nonorientable) of fixed genus. Furthermore, he claimed that for each orientable genus #>0 (respectively, nonorientable genus # >0, # {2) there is a complete graph of orientable genus # (respectively, nonorientable genus # ) and having connectivity attaining his bound. It is false that there is a complete graph of genus # (respectively, nonorientable genus # ), for every # (respectively # ) and that is the starting point of the present paper. Ringel and Youngs did show that for each #>0 (respectively, # >0, # {2) there is a complete graph K n which embeds in S # (respectively N # ) such that n is the chromatic number of surface S # (respectively, the chromatic number of surface N # ). One then easily observes that the connectivity of this K n attains the upper bound found by Cook. This leads us to define two kinds of connectivity bound for each orientable (or nonorientable) surface. We define the maximum connectivity } max of the orientable surface S # to be the maximum connectivity of any graph embeddable in the surface and the genus connectivity } gen(S # ) of the surface to be the maximum connectivity of any graph which genus embeds in the surface. For nonorientable surfaces, the bounds } max(N # ) and } gen(N # ) are defined similarly. In this paper we first study the uniqueness of graphs possessing connectivity } max(S # ) or } max(N # ). The remainder of the paper is devoted to the study of the spectrum of values of genera in the intervals [#(K n )+1, #(K n+1 )] and [#(K n )+1, #(K n+1 )] with respect to their genus and maximum connectivities. 1998 Academic Press
* Work supported by ONR Contract N00014-91-J-1142. Corresponding author. Work supported by NSF Grant DMS-9622780. Work supported by NSF Grant DMS-9622780.
208 0095-895698 25.00 Copyright 1998 by Academic Press All rights of reproduction in any form reserved.
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1. INTRODUCTION Throughout this paper, }(G) will denote the (vertex-)connectivity of the graph G. Similarly, denote by #(G) (respectively, #(G)) the orientable (resp. nonorientable) genus of G. For any real number x, denote by WxX the least integer greater than or equal to x and by wxx the greatest integer less than or equal to x. The maximum connectivity of a surface (orientable or nonorientable) is defined to be the maximum value of connectivity taken over all graphs embeddable in the surface. We denote this parameter by } max(S # ) when S # is orientable and by } max(N # ) when N # is nonorientable. It is well known that } max =5 for both the plane and the projective plane. In his 1973 paper, Cook [2] proved the following two results. Theorem 1.1. (a) if G is any nonplanar graph and if G is embedded in the orientable surface S # , then }(G)
\
5+- 1+48# , 2
while (b) if G has nonorientable genus at least 2 and if G is embedded in the nonorientable surface N # , then }(G)
\
5+- 1+24# . 2
Note that the bound in part (b) of the above theorem also holds for graphs having nonorientable genus 1. Cook then claimed that, by results of Ringel and Youngs [14] (resp. Ringel [12]; see also [13]), for each value of #>0 (resp., # >0, # {2), there is a complete graph K p embeddable in S # (resp. N # ) such that }(K p )=} max(S # ) (resp. }(K p )=} max(N # )). Therefore, } max(S # )=w(5+- 1+48#)2x and } max(N # )=w(5+- 1+24# )2x. Our first question is: Are these complete graphs the only graphs which attain Cook's bounds for } max(S # ) (resp. for } max(N # ))? It is well known that there are infinitely many 5-connected graphs which are planar and infinitely many 5-connected graphs with nonorientable genus 1. It has also been shown by Negami [810] that there are infinitely many 6-connected graphs embeddable in each closed surface with nonpositive Euler characteristic. On the other hand, it is also well known (see, for example, [16]) that any 7-connected graph embeddable in a surface of
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genus # or # has size bounded above by a function of the genus and hence there are at most only a finite number of 7-connected graphs embeddable in any surface, orientable or nonorientable. Thus for any surface with # 2 (resp. with # 3), the number of graphs embeddable in that surface and having connectivity }, 7 } } max , is finite. Furthermore, Cook also claimed that the complete graphs attaining his bounds for surfaces S # and N # are genus embeddable there. This is not necessarily the case. It is well known, in fact, that as genus increases, there are many surfaces which do not admit a genus embedding of any complete graph. The maximum connectivity of any graph having orientable (resp. nonorientable) genus # (resp. # ) will be denoted by } gen(S # ) (resp. } gen(N # )) and will be called the genus connectivity of the surface S # (resp. of surface N # ). Clearly, for any orientable surface S # (resp. nonorientable surface N # ) } gen(S # ) } max(S # ) and } gen(N # ) } max(N # ). But is there a surface with genus connectivity strictly less than its maximum connectivity? And precisely when does strict inequality hold in these inequalities? It is obvious that } max(S # ) and } max(N # ) are monotone nondecreasing functions of # and #, respectively. However, it will be shown that, surprisingly, when the genus increases, } gen may in fact decrease. For example, it will be shown later that } gen(S 35 )=23, but } gen(S 36 ) & = . 12 12 2 12 But then solving this inequality for r, we obtain r(n&6) 11 and since n&6>0, it follows that r 11(n&6). But then for n 18, r1 such that there exists a Class B surface in the interval [ g+1, g+x] (resp. in the interval [ g +1, g +x]).
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Let us begin our study of the breaking point by providing a lower bound. For all integers m 12, write m=12s+k, where s 1 and 0 k 11. Consider first the orientable case. Define # s, k =#(K m )=#(K 12s+k ). By the RingelYoungs genus formula [13, 14], # s, k =12s 2 +(2k&7) s+ W(k&3)(k&4)12X. Next, define $ k and = k as $k =
if k=1, 2, or 5 otherwise
&2,
{ &1,
and = k by
{
&1,
if
= k = 0, 1,
k=2,
if k=0, 1, 3, 5, or 6, otherwise.
Note that # s, k+1 &# s, k =2s+= k . To see this, note that the difference # s, k+1 &# s, k =12s 2 +(2k&5) s+W(k&2)(k&3)12X&(12s 2 +(2k&7) s+ W(k&3)(k&4)12)X)=2s+W(k&2)(k&3)12X W[(k&3)(k&4)12X. Upon computing the difference of the two ceiling functions, observe that # s, k+1 & # s, k =2s+= k , as claimed. We then have the following theorem. Theorem 3.1. Suppose that m=12s+k and that K m genus embeds in surface S #s, k so that } max(S #s, k )=} gen(S #s, k )=m&1. Suppose $ k is as defined above and suppose also that s+$ k 1. Then if i is an integer in the interval I s, k =[# s, k +1, # s, k +s+$ k ], S i is in Class B (and hence B{# s, k . It is also easy to check that for these same values, #(O(12s+k+1))=W f s, k(1)X . On the other hand, just as in the proof of Theorem 3.2, it may be shown that W f s, k(1)X g A . Finally, since #(O(m+1))># s, k , then by the definition of g A , g A #(O(m+1)). So assembling these inequalities, W f s, k(1)X g A #(O(12s+k+1))= W f s, k(1)X and the equality must therefore hold. K Consider now a possible extension of the preceding theorem to the case when s=0. Since #(K 1 )= } } } =#(K 4 )=0 and #(K 5 )=#(K 6 )=1, g A is not defined for k=1, 3, and 5. Suppose k=7. Then #(K 7 )=3 and #(K 5 )=4, so g A =4. But #(0(8))=3, so equality does not hold. Similarly, when k=11, #(K 11 )=10, #(K 12 )=12, and g A 11, but #(O(12))=10, so again equality does not hold. However, when k=9, #(K 9 )=5, #(K 10 )=7, and #(O(10))=6. Thus in this case #(O(10))= g A . Note that there are four of the 12 possible values of k which are not covered by Theorem 5.1 and there are six of the 12 possible values of k not covered by Theorem 5.2. Conjecture. In each of the 10 unsettled cases above, the breaking point is the orientable (nonorientable) genus of O(12s+k+1). In Section 2 we considered those surfaces in which the complete graphs genus embed and considered the problem of deciding when these complete graphs are the only graphs embeddable in these surfaces having connectivity } max for that surface. Of course, if we drop the demand that the complete graphs genus embed in a surface, we can look at the more general question as to when for any surface is a complete graph the only graph which embeds there and has connectivity } max . We now conclude by addressing this more general question. We begin with the orientable case.
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Recall that #(K 12s+k+1 )&#(K 12s+k )=2s+= k . Consider the surfaces S#s, k +r for 0 r 2s+= k &1. For all these surfaces, } max(S #s, k +r )=m&1= 12s+k&1. In the next theorem, it will be shown that for four of the 12 congruence classes modulo 12, one can completely decide when the graphs having connectivity } max are the only graphs embeddable in the corresponding surface. Theorem 5.3. If the surfaces S #s, k +r and the quantities $ k are defined as above, and if s+$ k 1, then (a) if 0 r s+$ k , K m is the unique graph with connectivity m&1 which embeds in S #s, k +r , while (b) if s+$ k +1 r 2s+= k &1, and k=1, 5, 7, or 11, then there is more than one graph with connectivity m&1 which embeds in S #s, k +r Proof. Suppose G{K m is a second graph which embeds in S #s, k +r and G has connectivity m&1. Then G must have at least m+1 vertices and so by inequality (3.1), #(G) f s, k(1)=#s, k +s+$ k +1 #s, k +r, a contradiction, and (a) is proved. To prove part (b), observe that, of course, Km is one graph which embeds in S#s, k and, hence, also in S#s, k +r for all r 0. By Theorem 2.2, #(O(12s+k+1)) =W((6s+(k&1)2)(6s+(k&5)2)3X and for k=1, 5, 7, and 11 and for these four values, the reader can easily show that this quantity, in turn, equals # s, k +s+$ k +1. So, for r=s+$ k +1 there is a second graph with }=m&1. K If s+$ k
