The set of orientable imbeddings of a graph can be partitioned according to the genus of the imbedding surfaces. A genus-respecting breakdown of the number of orientable imbeddings is obtained for every graph in each of two infinite classes. These are the first two infinite classes of graphs for which such calculations have been achieved, except for a few classes, such as trees and cycles, whose members have all their polygonal orientable imbeddings in the sphere

Furst, Merrick L.

Gross, Jonathan L.

Statman, Richard

Columbia University Academic Commons

GENUS DISTRIBUTIONS FOR TWO CLASSES OF GRAPHS Merrick L. Furst, Jonathan L. Gross & Richard Statman CUCS-193-85 3 0 0 c.Jo b er I q 8 S Genus Distributions for Two Classes of Graphs Merrick L. Furstl Jonathan L. Gross2 Richard Statman3 CUCS-193-85 The set of orient able imbeddings of a graph can be partitioned according to the genus of the imbedding surfaces. A genus-respecting breakdown of the number of orientable imbeddings is obtained for every graph in each of two infinite classes. These are the first two infinite classes of graphs for which such calculations have been achieved, except for a few classes, such as trees and cycles, whose members have all their polygonal orientable imbed dings in the sphere. lDepartment of Computer Science, Carnegie-Mellon University, Pittsburgh, Pa 15213. Support was provided by MCS-830805 and DCR-8352081. 2Department of Computer Science, Columbia University, New York, NY 10027. Most of the results herein were obtained while the second author was a visiting faculty member at Carnegie-Mellon University. His research was partially supported by ONR contract NOOOl4-85-K-0768. 3Department of Mathematics,Carnegie-Mellon University, Pittsburgh, Pa 15213. Sup-port was provided by MCS-8301558. 1 1 Introduction Furst and Gross [19851 have introduced a hierarchy of genus-respecting partitions of the set of imbeddings of a graph into a closed, oriented surface. This paper contains an illustration of a direct calculation of the genus distribution for every member of an infinite class of graphs called "closed-end ladders". It also contains an illustration of the use of a slightly finer partition in order to obtain the genus distribution for every member of another infinite class of graphs, which are called "cobblestone paths" . The choice of terminology here reflects the usual sensitivities of topo-logical graph theory. For instance, a graph may have multiple adjacencies _ or self-adjacencies. It is taken to be connected, unless one can readily infer otherwise from the immediate context. The closed orient able surface of genus i is denoted Si. An imbedding means a polygonal imbedding into a closed, oriented surface. Two imbed-dings into the same surface are equivalent if one can be continuously de-formed onto the other. When we say we are "counting the number of imbeddings," we are actually counting the number of equivalence classes of imbeddings. The size 01 a lace of an imbedding means the number of edge-traversals needed to complete a tour of the face boundary. If both orientations of the same edge appear on the boundary of the same face, then that edge is counted twice in a boundary tour. It is assumed that the reader is familiar with the elements of topology and graph theory, at the level of White [19731. However, we shall briefly review the relationship between rotation systems and graph imbeddings, which is described in Section 6.6 of White [1973] in slightly different termi-nology and somewhat reduced generality. A rotation at a vertex is a cyclic permutation of the edges incident on it, in which the two ends of a self-adjacency are considered separately. Thus, if a vertex has valence d, there are (d - I)! possible rotations there. A rotation system for a graph is an assignment of a rotation to each vertex. If a graph has vertices VI,"" Vn of respective valences d1 , ••• , dn , 2 then the total number of rotation systems is n II (c4 - I)! i=l A research abstract of Edmonds [1960] called explicit attention to a bijective correspondence between the set of imbeddings of a graph G and the set of rotation systems. (The correspondence seems to be implicit in the pioneering work of Heffter [1891].) It follows that the total number of imbeddings of a graph is the same as its number of rotation systems. Details for the simplicial case (i.e. without self-adjacencies on multiple adjacencies) were first giyen by Youngs [1963]. A generalization to the non-simplicial case was developed by Gross and Alpert [1974]. The bijective correspondence is realized if one considers the permutation action of the rotation system on the set of oriented edges. IT e is an oriented edge from vertex u to vertex 11, this action permutes e to the reverse of the oriented edge that follows e in the rotation at 11. The edge-orbits of this action are taken to be the face-boundaries of an imbedding. 3 2 Closed-end ladders Imagine that rounded pieces of material are used to close both ends of an n-rung ladder. A mathematical model of this object may be obtained by taking the graphical cartesian product of the n-vertex path P,. with the complete graph K2 and then doubling both its end edges. We call the resulting graph an n-rung closed-end ladder and denote it Ln herein. Figure 2.1 depicts a closed-end ladder. (1 1 1) Figure 2.1 The 3-rung closed-end ladder Ls. The horizontal edges are said to form "sides" of the ladder. The two curved edges are called "ends" or "end-rungs". Ladder-like graphs played an extensive role in the solution by Ringel and Youngs [1968] to the Heawood Map-Coloring Problem (see Ringel [1974]). In fact, we shall use the picture method of Gustin [1963], so important to that solution, to specify every rotation system - and accordingly, every imbedding - of a ladder graph. We note that a trivalent vertex has only two rotations. IT the vertex is drawn solid, the rotation is counterclockwise. It the vertex is drawn hollow, then the rotation is clockwise. Figure 2.2 shows a rotation system for a 4-rung ladder and its two edge-orbits, one dotted, the other dashed. The graph L" has 8 vertices and 12 edges. The imbedding depicted has two faces (one for each edge-orbit). Substitutions on the right side of the 4 ( \ """ . ---, .-----'''''---, or- ~'\ ... I I ---- --i r'. I I \ t r : ! o '0 ..... ,---~_./ '-_ .. -----.--.. - ...... ----- .... -- .. -------4.-... -....... Figure 2.2 A rotation system for the 4-rung ladder L .. and its two associated edge-orbits. Euler polyhedral equation 2-2"'(= V -E+F yields the equation 2 - 2"'( = 8 -12 + 2 = -2 from which we infer that the imbedding surface associated with Figure 2.2 has genus "'( = 2. If both endpoints of a rung are solid, or if both are hollow, or if the rung is an end-rung, we call it a matched rung (in the given rotation sys-tem). Otherwise we call it an unmatched rung. Thus, Figure 2.2 has three matched rungs. Two matched rungs are said to be consecutive matched rungs if no other matched rungs lie between them. Two consecutive matched rungs are said to be evenly separated if the number of interposing unmatched rungs is even (including zero). Thus, the left end-rung of Figure 2.2 is evenly separated from the doubly hollow matched rung, but the doubly hollow matched 5 rung is oddly separated from the right end-rung. Thus, the number of edge-orbits (two) is one more than the number of evenly separated pairs (one) of consecutive matched rungs. We generalize this observation about Figure 2.2. Lemma 2.1 The number of edge-orbits induced by a rotation system for a closed-end ladder Ln equals one plus the number of evenly separated pairs of consecutive matched rungs. Proof: Suppose that the total number of matched rungs is m. Let us begin by considering any rotation system of the ladder Lm such that every rung is matched. It is not difficult to verify that such a rotation system has m + 1 edge-orbits, and that three different edge-orbits are incident on each vertex. (The aid of a few drawings is highly recommended.) The rest of this proof is concerned with the effect of inserting a string of unmatched rungs between two matched rungs. Tracing the orbit lines in Figure 2.3 is sufficient to demonstrate that whenever a 2-string of similar unmatched rungs is inserted between two arbitrary rungs, there is no effect on the number of edge-orbits. It follows that when we insert strings of unmatched rungs into the lad-der Lm , we may as well assume that consecutive unmatched rungs are dissimilar. Let's call this an alternating string of unmatched rungs. Tracing the edge-orbits in Figure 2.4 indicates tl1at inserting an alter-nating 3-string of unmatched rungs between any two kinds of rungs has the same effect as inserting only the middle rung of the string. By combining the observation about alternating 3-strings with the ob-servation about 2-strings of similar matched rungs, we may infer that the effect of inserting any odd-length string of unmatched rungs is the same as inserting one unmatched rung. Similarly, we may infer that the effect of in-serting any even-length string of unmatched rungs is the same as inserting either a 2-string of dissimilar unmatched rungs or no rungs at all. In order to insert a single unmatched rung between two matched rungs, we proceed in two stages. First, we insert a matched rung, which increases the number of edge-orbits by one. We observe that each endpoint of the new rung is incident on three distinct edge-orbits. Thus, when the rotation at one end of the new rung is reversed (i.e. this is stage two), its three 6 ----------- ~-------- ... -]----~-__ L __ --1 __ ----------- ---------Figure 2.3 The two possible 2-strings of similar matched rungs and their local edge-orbit structure. edge-orbits become one orbit, for a reduction by two. The net effect of inserting the unmatched rung is a decrease of one edge-orbit. Another edge-tracing argument confirms that inserting an alternating pair of unmatched rungs between two consecutive matched rungs causes no net change in the number of edge orbits. QED Lemma 2.1 enables u.s to complete the derivation of the genus distri-bution of ladders by straightfoward enumerative techniques. We employ two auxiliary expressions in what follows. One is 8 (n, m, k), which stands for the number of rotation systems for the ladder Ln that have m mathed non-end rungs, of which k pairs are evenly separated. The other is b(p, q, r), which stands for the number of ways to put p identical balls into q distinct boxes, so that exactly r boxes have an even number of balls. To obtain a combinatorial expression for b(p, q, r), we imagine that one ball is placed into each of the q - r odd boxes and that the remaining p-q+r 7 ---- ,.-----------, ---3:' 1 ----: r---- ... --I 1- ... ----1 :-----. I I I I , ~ ) ,----------, - ... I '" , 1 '---r 'j ( I I , , ____ .J '-____ ~ _---J L ____ \ _ J 1_ , 2 ------ '\----_ .. _----, - --~"\ ,-.. 4 ' , 2 - '- ________ - __ I _ Figure 2.4 The equivalence between inserting an alternating 3-string of unmatched rungs and inserting only the middle rung of the string. balls are then distributed in pairs into the q boxes. Thus, { 0 if p - q + r is odd, b(p, q, r) = (q). ((p-Q+r)fHq-l) otherwise Q-r Q-l or, equivalently, . { 0 if p - q - r is odd, b(p, q, r) = (:) . ((P-Q+;~;+Q-l) otherwise (1) In order to analyze s(n, m, k) we imagine that the n - m unmatched non-end rungs are to be inserted into the m + 1 distinct boxes formed along the ladder L,. by the m matched rungs, so that the k boxes are filled with evenly many unmatched rungs. Clearly we have s{n, m, k) = 2"b(n - m, m + 1, k) (2) If n = k (mod 2) , then (n - m) - (m + 1) + k is odd, from which it follows that s( m, n, k) = O. However, if n ~ k (mod 2), then we combine 8 I I equations 1 and 2 to obtain (3) We now define f(n, k) to be the number of imbeddings of the ladder graph Ln that have k faces. According to Lemma 2.1, we have n f(n,k) = L s(n,m,k -1) (4) m;O Using equation 3, we transform equation 4 into f(n,k) =2n t (m~k) ((n+k-2)/2) m=O k 1 m (5) or, equivalently f(n,k) = 2n t (m~k) ((n + k)/2 -1) m=O k 1 m (6) Using the combinatorial identity (7) we convert equation 6 into which separates, in turn, into the form f(n, k) = 2n t ( : ) ((n + k)/2 - 1) +2n t ( : ) ((n + k)/2 - 1) m=O k 1 m m=O k 2 m (9) The combinatorial identity 9 enables us to determine from equation 9 that Therefore, we may infer We now reuse the combinatorial identity 7 to obtain The combinatorial identity implies that (p - 1) (p) q q-l=·qp ( (n + k)/2 - 1) = ((n + k)/2) 2(k - 1) k - 2 k - 1 (n + k) This allows us to simplify equation 12 to conclude f(n, k) = 2(3n-k)/2 ((n + k)/2) (1 + 2k - 2) k-l n+k whenever n == k (mod 2). Otherwise, f(n,k) = o. (13) In order to convert the face-count formula in equation 13 into a genus distribution formula, we use the Euler polyhedral equation in the form 2 - 2i = #V{Ln) - #E{Ln) + #F(Ln -+ Si) = 2n - 3n + k Thus, when the genus of the imbedding surface is equal to the number i, the number of faces is k = n+ 2 - 2i 10 Let g,(Ln) denote the number of imbeddings of the ladder Ln in the surface SI. It follows that gi(Ln) = f(n, n + 2 - 2i) (14) When we apply equation 13 to the right-side of equation 14, we obtain the equation o(L ) = [3n-(n+2-2i)J/2 ([n + (n + 2 - 2i)]/2) ( 2(n + 2 - 2i) - 2) g. n 2 (0) 1 + ( 0) . n + 2 - 2, - 1 n + n + 2 - 2, This simplifies routinely to o(L ) = 2n-1+, (n + 1 - i) 2n + 2 - 3i g. n 1 2· 1· n+ -, n+ -I and, since (a~.) = (:), we have o(L ) = 2n-1+i(n+ 1- i) 2n + 2 - 3i g. n • + 1 ° I n - I (15) The following table shows the genus distribution for small values of nand I. go gl g2 gs g. Ll 2 2 0 0 0 ~ 4 12 0 0 0 Ls 8 40 16 0 0 L. 16 112 128 0 0 L5 32 288 576 128 0 11 3 Cobblestone paths Suppose that every edge of the n-vertex path Pn is doubled, and that a self-adjacency is then added at each end. Figure 3.1 shows how the resulting graph might be drawn. It seems appropriate to dub this graph a cobblestone path We denote it I n herein. Figure 3.1 The cobblestone path J •. For any connected graph G, and for i = 0,1, ... , let gi(G) be the number of imbeddings of G into the closed orient able surface Si. We may regard the gentU distribution for G as a vector Obviously, only finitely many entries are non-zero. Our objective is to calculate all the numbers gi(Jn ), for i = 0,1, ... , and for n = 1,2 .... Sometimes we abbreviate gi(Jn ) by gi,n herein. 12 The recursion construction assures that we have a cobblestone path I n - l positioned horizontally, as in Figure 3.1. The subsequent cobblestone path I n is obtained by first imposing a new vertex at the middle of the right end loop and then attaching a new right end loop at the new vertex. To establish a recursion formula, it is necessary to distinguish between two kinds of imbed dings of the cobblestone path I n - l , depending on whether the two occurrences of the right end edge lie on two distinct faces or on the same face. For i = 0, 1, ... and for n = 1,2, ... we define d; (J n), sometimes abbreviated d;,n , to be the number of imbeddings of I n in Si such that the two occurrences of the right-end edge lie on distinct faces, and we define Si(Jn ), sometimes abbreviated Si,n , to be the number of imbeddings of I n in Si such that the two occurrences of the right-end edge both lie on the same face. Obviously, we have the equation For each cobblestone path I n , we may form vectors The basis step for the recursion is the following observation (SO,l! Sl,ll S2 ,1,''') = (0,2,0, ... ) (16) (17) (18) In constructing the cobblestone path I n from its predecessor I n - 1 , we are adding a new vertex of valence 4. Since (4 - I)! = 6, it follows that I n has six times as many imbeddings as I n - 1• In fact, the cobblestone path I n has 6n imbeddings, for n = 1,2, .... Our viewpoint is that each individual imbedding of I n - 1 gives rise to six imbeddings of I n , which occurs by way of the intermediate graph J;_l' By J;_l we mean the result of inserting a new vertex at the midpoint of this right-end loop of the cobblestone path I n - 1 • The six dashed arcs in Figure 3.2 illustrate the six ways the new right-end loop for I n can be attached at the new vertex of I n - 1 • Now consider any imbedding into the surface Si of the cobblestone path I n - 1• If both occurrences of the right-end loop of I n - 1 are on the same face, 13 -- ...... I..... --". - --_ _ L \) _- ... I -_, + _- . i -- -- --': e -- \~ , __ >fo ___ I , ...------ / "------ : '. __ (,1'~ -__ I ' .... ____ ~..- _ '- J -.,., Figure 3.2 The six ways of attaching a new right-end loop. then every one of the six ways of attaching a new right-end loop can be realized in the surface Si, that is, without attaching an extra handle to Si. Obviously, the two occurrences of the new right-end loop appear on different faces of the resulting imbedding of J". However, if the two occurrences of the right-end loop of J,,-l are on different faces of its imbedding in Si, then only the four monogon-generating insertions of the new loop are in Si. The two insertions in which the new right-end loops runs from one face to another require the addition of a handle from one face to the other. In this case, both occurrences of the new right-end loop lie on the same face of the new imbedding. Thus, we have established the simultaneous recursion formulae c4(J,,) = 4c4(Jn-l) + 6si (Jn - 1) Si(Jn ) = 2c4-1(Jn- 1) 14 (19) (20) The solution of the recurrence begins with a substitution of 2c4-I(Jn - 2) for si(Jn-d into equation 20 which yields the simplified re-currence relation (21) By reversing the recursion, we may calculate values (22) This artifice enables us to define 00 Di(X) = Ec4(Jn)xn i=O in preparation for an infinite summation on equation 21, as follows. 00 00 00 E c4,nxn = 4 L c4,n_IXn + 12 E c4_I,n_2 Xn n=2 n=2 n=2 00 00. = 4x E c4,n_IXn- 1 + 12x2 E lit_I,n_2 Xn- 2 n=2 n=2 Therefore, and, consequently Di(X)(1 - 4x) = 12x2 Di-I(X) = c4,1 x + c4,o (23) From equations 17 and 22, we know that lit,l = 0 and lit,o = 0, for all i ~ 1. Thus we may simplify equation 23 to the linear recursion Di(X) = ( 12x2 ) Di-I(X) , for i ~ 1 1-4x (24) We will now proceed to establish the value of the polynomial Do{x). From the Jordan curve theorem, we know that SO,n = 0, for n ~ 1. Accord-ingly, we may conclude from equation 19 that 15 Since do.o = 1, we infer that 1 Do(x) = 1- 4x We easily combine equations 24 and 25, to obtain the result (12x2)i Di(X) = (1 _ 4x)i+l The coefficient of xr in the power series expansion of (1 - axt' is (25) (26) (For instance, see Tucker [1980, p. 84J or Liu [1968, p. 31J.) It follows that the coefficient of x"-2i in the power series expansion of (1 - 4X)-(i+l) is ((n - 2i) + (i:- 1) -1)4n - 2i = (n - i.)4n - 2i n - 2& n - 2, Thus, the coefficient of x" in the power series expansion of Di(X) is ( .) (.) (0) o ° n-, ° ° n-, ° ° n-t 12' . 4n - 2, • 0 = 3' ·4n -, • 0 = 31 ·4n - 1 • ° n - 2, n - 2, , Thus, c4(Jn ) =3i ·4n - i • (n~'),fori~O and n~O (27) We now recall equation 20 Si(Jn ) = 2c4-1(Jn -d, for i ~ 1 and n ~ 1 and infer that Si(Jn ) = 2. 3i - 1 • 4n - i . (; ~:) (28) Therefore, from equation 16, we conclude g,(Jn) = 3i '4n - i .(n ~ i)+2.3i - 1.4n- i .(; ~:) for i ~ 0 and n ~ 1 (29) The following table contains some of the small values. 16 90 91 92 Total J1 4 2 0 6 J 2 16 20 0 36 J s 64 128 24 216 J. 256 704 336 1296 17 REFERENCES 1. J. Edmonds [1960], A combinatorial representation for polyhedral surfaces, abstract in Notices Amer. Math. Soc. 7, 646. 2. M. Furst and J. L. Gross [1985], Hierarchy for imbedding-distribution invariants of a graph. 3. J.L. Gross and S.R. Alpert [1974], The topological theory of current graphs, J. Combinatorial Theory 17, 218-233. 4. W. Gustin [1963], Orientable embedding of Cayley graphs, Bull. Amer. Math. Soc. 69, 272 -275. 5. L. Heffter [1891], Uber das Problem der Nachbargebiete, Math. Ann. 38, 477-508. 6. C.L. Liu [1968], Introduction to Combinatorial Mathematics, McGraw - Hill. 7. G. Ringel [1974], Map Color Theorem, Springer-Verlag. 8. G. Ringel and J.W.T. Youngs [1968], Solution of the Heawood map-coloring problem, Proc. Nat. A cad. Sci. USA 60, 438-445. 9. A. Tucker [1980], Applied Combinatorics, Wiley. 10. A.T. White [1973], Graphs, groups and surfaces, North-Holland. 11. J.W.T. Youngs [1963], Minimal imbeddings and the genus of a graph, J. Math. Mech. 12, 303-315. 18 

Genus Distributions for Two Classes of Graphs

https://academiccommons.columbia.edu/doi/10.7916/D8NZ8GK1/download

Genus Distributions for Two Classes of Graphs

Abstract

Similar works

Full text

Available Versions

Columbia University Academic Commons