Vol. 4 No. 2 ACTA MATHEMATICAE APPLICATAE SINICA May, 1988 .... H,

THE CONNECTIVITY OF Z-TRANSFORMATION GRAPHS OF PERFECT MATCHINGS

OF HEXAGONAL SYSTEMS

(.~ i~jia~g University)

(Col~ese of Finance and Economics, ]Fuzhou University)

A b s t r a c t

Let H be a hexagonal system. The Z-transformation graph Z(H) is a graph where the vertices are perfec~ matchings of H and where two perfect matchings are joined by an edge provided their ~yn~megric difference consists of six edges of a hexagon of H. We prove that the connectivity of Z(/:/) is equal to the minimum degree of vertices of Z (H).

A hexagonal system H , also called " h e x a n i m a l " or "honeycomb system" (see, e.g., [ 1 ] ) i s a finite connected plane graph with no cut vertices in which every inter ior region is bounded by a regular hexagon of side length i m~. Graphs of this kind are of chemical significance since a hexagonal system with at least one perfect matching is the skeleton of a benzenoid hydrocarbon molecule. Recall that a perfect matching of a graph G is a set of disjoint edges of G covering all the vertices of G.

The Z- t ransformat ion graph Z ( E ) of a hexagonal system ~ is defined C~] to be a graph where the vertices are the perfect matchings of ~ and where two perfect matchings are joined by an edge provided ~heir symmetr ic difference consisLs of s~x edges of a hexagon of H (see Fig. 1).

(

I Fig. 1 A hexagonal system H and the Z-transformation graph Z (H)

Received August 1, 19~6. " P ~ j e c t supporte~l by the National Natural Science Foundation of China.

132 ACTA MATHEMATICAE APPLICATAE SINICA Vol. 4

In order $o simplify the discussion, a hexagonal system S is to be placed on a plane so that two edges of each hexagon are ver$ical. Let M be a perfect ma~ching of / t . An M-alterna$ing hexagon of H, i.e., a hexagon whose six vertices are covered by three edges of M, is called, separately, as proper and improper sextet m as shown

© © proper sex%e$ improper sextet

~.

in F~g. 2. Evidently, for any perfect matching M of a hexa-

gonal system H, the degree d(M) of M in Z ( H ) is equal to the number of all M-alternating hexagons, i.e., the number of proper sextets of M plus the

number of improper sextets of M. The sexte$ rota$ion m is defined to be a simultaneous roSa$ion of all the proper sexSets of a given perfect matching M~ of ~" into ~he improper s e x ~ to give another perfect ma$ch~g of .H. The author of [5] proved that for any perfect matching of ~R', after repeating the sextet rotation for a sufficient number of times, we shall necessarily arrive at the root Kekule pastern, i.e., the unique perfect matching which has no proper sexSets c~. From the fac$ tha$ the sextet ro~$ion can be regarded as a series of rotations of a s~ngle proper sextet, we deduce that there is a path in Z(J~) connecting each perfect matching and the root Kekule pattern. Therefore, Z (H) is connected. In the present paper we consider the connectivity of Z (H) .

Le$ M be a perfect matching of a hexagonal sys tem/~ and 8 bean M-alternating hexagon of H. We use /f~ to denote t~e s~me~ric difference of M and the set of edges of 8, i.e., M A ~ ( M U E ( 8 ) ) - (MN E(8)) . Let ~ and # be two M-alternating hexagons having an edge in common. An Fig. 3 A maximal M-chain with respect to (~, g)

F t jl~-chain with respect to (8, 8') is a series of hexagons ~r, "", zl, #, 8, ~1, "". ~t which cons$imSes a hexagonal system and whose censers lie in a straight line and each of which except 8 and 8' contains exac~y Swo edges in M. Let L be an M-chain with respect to (8, 8'). L is called a maximal M-chain with respect to (8, s') ff L is not a proper subchain of any M-chain with respect to (8, s').

We need the following lemma. L e m m a 1. LeA H be a hexagonal system with perfect matehings, and let M1

and M~ be perfect ma$chings of G. The distance d(~/1, ~/~) between M1 and M~ in Z ( H ) is two. Then there are 8 ( Z ( H) ) internally disjoint paths joining M~ and M~ in Z ( H ) , where 8(Z(]~)) is the minimum degree of vertices of Z ( H ) .

Proof. Let ~M~M'M~ be a 2-paSh joi~ing M1 and M~, where M'-MA81, M~ • -M'zts2, ~ and 82 are M1 and M'-alterna:~ing hexagons, respectively.

Denote by A the set of all Ml-alterna$ing hexagons except 8~, 8~ (if 82 is an M1- alternating hexagon), and B the subset of A in which each hexagon has edges in common with neither 81 nor 8~. I$ is not difficult $o see that for each &E B, there is an associated path, denoted by p(8~), connecting M1 and M~: T(8~)=M1M~M~,MoMa, where M 6 - M~As~, M~ ~ M~A~, M~-- M~As~, and M~ ~ M~A~. We denote

P(B) = {p(~)ts,~B}.

Obviously, the paths in P(B) are pairwise inSernally dNoin$.

No. 2 CONNECTIVITY OF Z-TRANSFORMATION GRAPHS 133

!n the following, without loss of generality, we may assume that d (S/~) ~<d (M~). The following %we cases are considered.

Case 1. E (~ ) ~E(s=) 4=¢. In this case s, is not an Mx-al~erna¢ing hexagon. Thus d(M~) = IAI +1. IfA=.B,

then d (M~) = [B I + 1. The paths of P (B) together with the path lo which is indicated at the beginning of the proof form d(M~)>~8(Z(H)) internally disjoin% paths as desired. Now suppose A - B @ ~.

Subcase 1.1. There exists x ~ A - B such that E(za) NE(sx) @(2J as shown in Fig. 4.

By the assumption d(M D <d(MD, ~here mus~ be an S/~-alternating hexagon

which has an edge in common with s~ as depicted in Fig. 4. We deduce ~ha~

F%.4

A - B = {xa}. Hence d(s/x)--IAt + I - I B I + 2 Let £~,---, z[, x~, sl, xl=s~, z2=x~,---, be the maximal S/x-chain with respect to (sx, ~ ) . Then the following perfect

matching M" of H satisfying d (Mx) - l :~d (M') ~ 8 (Z (H) ) is

M" = ( ( . . . ( ( ( i ~ A s z ) As=) zl~'~) A...A~,_x) Ax,) .

Now the paths of P(B) Ui~ are the desired d(M~) - I > 8 ( Z ( H ) ) internally disjoin* paths.

Suboase 1.2. Any M~-alternating hexagon X~ E A - B has no edge in common with s~. Then we have thatE(X~)NE(s~)@O, and IA--B[~2.

Subsubcase 1.2.1. There exists exactly one Mx-al~rna¢ing hexagon x, having an edge in common with s~. By the assumption that d(s/x)~d(M~), there must bean S/~-al~ernating hexagon ~ which has an edge in common w i ~ sx (see Fig. 5), ~ is either x* or x~.

Now we find a path io* joining Mx and S/~: p*= MxM~M,M~M,M~S/~, where M, -- M~A~, M~ = M,~, M~ = M~x'~, M, = M~Ax~, S/~ = M.As~ and M~ = M~Az'~. It is

not difficul~ %0 check that the paths o fP(B) U {P, ~*} are d(M~)= IBI +2 internally disjoint paths joining M~ and S/~ in Z (H) as desired.

Ft. 5 ~=x*. or x'~=~'* Fig.

Subsubcase 1.2.2. There are two hexagons x~ and x~ E A - B such that E ( ~ ) .E(s~) --/=@, E(x~) ['] E(s~) --/=0. Again by d(M~) <~d(M~), there are two M~-alternating hexagons ~ and ~ each of which has a common edge with sx (see Fig. 6).

As before we find two patt~ joining Mx and M~ as follows:

where

134 ACTA MATHEMATIOAE APPLIUATAE SINICA Vol. 4

and

M',= M ' = M ' - - M"

We can verify that the paths of P(B) U {20, p*, p*'} are d(M D =- IBI ~-S internally disjoint paths joining Mz and Mj in Z ( H ) as desired.

Case 2. E(81) FI.B(82) =~. When there are t ~ 2 hexagons in A - B each of which B~ one edge in common

with 81, any M=-alternating hexagon (@~1) has no edge in common with 81 (see Fig. ~ ~ / 7). By the assumption that d(M~) ~d(M~), there exist at least t M~-

alternating hexagons each of which has one edge in common with 82 and therefore any one in A - B h~q no edge in common with 8~. Hence IA-BI = t and d(M1)--IB[ +t+2. Then the pathg of P(B) ~ogether with p ( indica~l at the b e g ~ g of the proof) and p'=M1M"M=,

Fig. 7 where M"=MI~Js~, form [B I ~-2 internally disjoin~ paths joining M~ and M~ in Z(H). The remainder is ~o check ]B I .+.2=d(M~) - t~8(Z(H)) . In fact, the perfect mash ing M*=M~A~ has degree d(M') =d(Mx)-t.

Now suppose that there is exactly one hexagon ~ in A-- B such tha~ E ( ~ ) FI E (~ ) @~. If A-B= {~}, then d(M~) = ]B I +3. An analogous argument as in Subcase 1.1 will show that there is a perfect matching M* such that d(Mx) - l~d (M*) ~ 8 ( Z (H) ) . Now the conclusion follows from the fac~ that the paths of P (B) ~ogether with ~p and ~'--MzM"M~, where M " = M ~ , form ]B I + 2 = d(M1) - 1 9 d ( M ' ) ~ 8 (Z(H)) in~rna l ly d/sjoint paths joining M~ and M~ in Z (H) . Then we have to show ~ha~ there is at least one Mx-alterna~ing hexagon differen~ from ~ , say ~/~, which has an edge in common with $~. The assumption d(Mx)~d(M~) ensures that each of

r~ .s

No. 2 CONNECTIVITY OF Z-TRANSFORMATION GRAPHS 135

and s~ must have an edge in common with an M~-a]ternating hexagon. As before we .can find two maximal Ml-chains L1 and L~ with respect to (sl, x~) and (s2, y~), respectively. If L1 and L~ have edges in common, there are oniy two types (up to a symmetry) as shown fix Fig. 8. No matter whether Lx and L2 have edges in commo]l or not, there is a perfect matching M ° as follows:

S/* = (... (( (... ((Ml~x~) ~zl) A"" A~) ~8~) Ayl) ~"" ~ ) .

Evidently, d(M'~ ~d(M1) - 2 . Then the paths in _P(B) ~oge%her wi~h p, p' (the paths and T' are defined as in the preceding paragraph) are d(M:t) -2>~d (M*)>~8 (Z (H))

in~ernally disjoin~ paihs joining M~ and M2 in Z(H) as required. The proof is ~hus completed. T h e o r e m 2. The connectivity of Z(H) is equal to 8 ( Z ( H ) ) , where 8 (Z (H) )

is the minimum degree of vertices of Z(H). Pvoof. As previously mentioned, Z(H) is connected. Let G be a cut se~ of

Z(H) . By the above lemma, we can easily deduce that tGt ~>8(Z(H)). Then the theorem follows.

References .[ 1 ] I~arary, F., The aell growth problem and its attempted solu~ioas, Beitr~ge zur Graphentheorie (Int.

Ko]l. Manebach, 9--12. Mai 1967), ed. H. Sachs, H. - - J . Voss, and H. Walther. B. G. Teubner Veriagsgesellschaft Leipzig 1968, 49--60.

[ 2 ] Zhang Fuji, Chen Rongsi and Guo Xiaofeng, Perfect Matchings in Hexagonal Systems, Graph and Comb/natoc/cs, 1 (1985), 383--386.

.[ 3 ] ghang Fuji, Guo Xiaofeng and C"aen l~ongsi, Z-transformation graphs of perfect matchings of hexagonal systems, Proceeding of the First Japan laternational Conference on Graph Theory and Applications.

[ 4 ] N. Ohkami, A. Motoyama, T. Yamaguchi, H. Hosoya and I. Gutman, Graph-theoretical analysis of the Clar's azomatic sextet, T~rahed~'on, 37 (1981), 1113--1122.

.[ 5 ] Chen Zhibo, Directed tree structure of t]ae set of KekuI6 pattern~ of polyhex graphs, Chem~ca~ ~hy~i~ Letter&, 115 (1985), 291--293.