Relations between Clar structures, Clar covers, and the sextet-rotation tree of a hexagonal system

Transcription

1 Discrete Applied Matematics 156 (2008) Relations between Clar structures, Clar covers, and te sextet-rotation tree of a exagonal system San Zou a, Heping Zang a,, Ivan Gutman b a Scool of Matematics and Statistics, Lanzou University, Lanzou, Gansu , PR Cina b Faculty of Science, P.O. Box 60, Kragujevac, Serbia Received 14 Marc 2006; received in revised form 16 August 2007; accepted 26 August 2007 Available online 25 October 2007 Abstract Sextet rotations of te perfect matcings of a exagonal system H are represented by te sextet-rotation-tree R(H), a directed tree wit one root. In tis article we find a one-to-one correspondence between te non-leaves of R(H) and te Clar covers of H, witout alternating exagons. Accordingly, te number (nl) of non-leaves of R(H) is not less tan te number (cs) of Clar structures of H. We obtain some simple necessary and sufficient conditions, and a criterion for cs = nl, tat are useful for te calculation of Clar polynomials. A procedure for constructing exagonal systems wit cs < nl is provided in terms of normal additions of exagons Elsevier B.V. All rigts reserved. Keywords: Hexagonal system; Perfect matcing; Clar cover; Clar structure; Sextet-rotation-tree 1. Introduction A exagonal system is a connected plane grap witout cut vertices, in wic eac interior face is a regular exagon of side of lengt one [16]. In tis paper we are interested in exagonal systems tat possess perfect matcings. A perfect matcing of a grap H is a set of pairwise disjoint edges tat cover all vertices of H. One sould note tat te carbon-atom skeleton of a benzenoid ydrocarbon is a exagonal system [6]. Terefore exagonal systems and teir matematical properties were muc studied in cemistry. In cemistry instead of perfect matcings one speaks of Kekulé structures and te edges contained in a perfect matcing are referred to as te double bonds of te respective Kekulé structure. Kekulé structures ave numerous applications in cemistry [6]. For instance, various Kekulé-structure-related models for approximating te Dewar resonance energy (DRE) [17] of benzenoid ydrocarbons ave been proposed, suc as te Swinborne-Seldrake [20], te Herndon Hosoya [11], etc. In te Swinborne-Seldrake model, DRE is expressed in terms of te number of Kekulé structures. Eventually an improved formula for DRE was put forward [9], based in te sextet-rotation-tree. Te sextet rotation, transforming all proper sextets of a Kekulé structure into improper sextets, results in a directed tree wit one root [13,1]; in wat follows tis tree, pertaining to a exagonal system H, is referred to as te Tis work is supported by NSFC Grant no Corresponding autor. addresses: zous@lzu.edu.cn (S. Zou), zangp@lzu.edu.cn (H. Zang), gutman@kg.ac.yu (I. Gutman) X/$ - see front matter 2007 Elsevier B.V. All rigts reserved. doi: /j.dam

2 1810 S. Zou et al. / Discrete Applied Matematics 156 (2008) sextet-rotation-tree and is denoted by R(H); details of its construction are given below. Analogously, counter-sextet rotation also produces a directed tree R c (H ), wic, in te general case, needs not be isomorpic wit R(H). Onte oter and, R(H) and R c (H ) ave te same eigt and widt (te number of leaves, vertices of in-degree 0). Tis remarkable property was first observed by Gutman et al. in [8] and later verified by Zang et al. [25] in a more extensive sense. Hence nl(h ), te number of non-leaves in R(H), is an invariant. In [9] te formula DRE(H ) = ln K(H) ln nl(h ) (1) was deduced, were K(H) denotes te number of Kekulé structures of H. In view of tese cemical applications, it is purposeful to classify te Kekulé structures into leaves and nonleaves, according to te structure of te sextet-rotation-tree. A spanning subgrap of H is called a Clar cover [26] if eac of its components is eiter a exagon or K 2.An alternating exagon of a Clar cover of H is a exagon of H wose edges belong alternately to te edge set of te Clar cover and its component (wit respect to te edge set of H). In tis article we first establis a one-to-one correspondence between te non-leaves of R(H) and te Clar covers of H, witout alternating exagons. Hence nl(h ) = cc(h ), were cc(h ) denotes te number of Clar covers witout alternating exagons. In te Herndon Hosoya s model, te concept of (generalized) Clar structure was introduced, see below. In connection wit tis, El-Basil and Randić [15,14,3] conceived te Clar polynomial, te counting polynomial of Clar structures (in terms of te number of exagons tey contain), and described various approaces for its computation. Based on te concept of Clar cover, a more precise grap-teoretical definition of Clar structure could be given [18]: A Clar cover of H is called a Clar structure if te set of exagons is maximal (in te sense of set-inclusion) witin all Clar covers of H. 1 Hence cs(h ) cc(h ) = nl(h ), were cs(h ) is te number of Clar structures of H. Te Clar polynomial [3] of a exagonal system H can be defined as ρ(x, H ) = ρ(i, H )x i i 0 (2) wit ρ(i, H ) denoting te number of Clar structures of H wit i circles (or exagons). Actually, Gutman [5] stated tat every perfect matcing of a exagonal system contains tree edges of a exagon. Ten te index i may start from 1 as tere are no Clar structures wit zero exagons. Clearly, if cs(h ) = cc(h ), ten te problem of computing Clar polynomial is somewat less difficult, since it can be solved by constructing all Clar covers witout alternating exagons. In order to caracterize te exagonal systems wit cs(h ) = nl(h ), in Section 3 we recall a classical result of Zang and Cen [21]: For a exagonal system H, r(h) K(H), and equality olds if and only if H contains no coronene (see Fig. 3) as its nice subgrap. Here r(h) denotes te number of sextet patterns in H, a set of exagons in a Clar cover. Below we provide a simpler proof of tis result, using te concepts of cut lines and g-cut lines. In Section 4 tis approac is used to find a simple sufficient condition for cs(h )=cc(h ): If a exagonal system H as no coronene as its nice subgrap, ten cs(h )=nl(h ). Te converse of tis statement does not old, and in te sequel we deduce a general necessary and sufficient criterion. Various examples of exagonal systems wit cs(h ) = nl(h ) are constructed and teir Clar polynomials are computed. Finally a construction procedure for exagonal systems wit cs < nl is provided in terms of normal additions of exagons. 2. Identity cc(h ) = nl(h ) For convenience, any exagonal system H considered, is assumed to be placed in te plane so tat one of its edgedirections is vertical. Te peaks and valleys of H (see [6]) are colored black and wite, respectively. In wat follows, all cycles considered are assumed to be oriented clockwise. Tis convention will play an important role in te following considerations. Let H be a exagonal system wit a perfect matcing M. A cycle C of H is said to be M-alternating if its edges belong alternately in M and E(H)\M. A pat P is M-alternating if every inner vertex of P is incident wit an edge in P M, but te end edges of P are not in M.AnM-alternating cycle C of H is said to be proper if eac edge of C belonging to 1 Anoter non-equivalent definition, muc used in te cemical literature [6], requires tat te number of exagons be maximal.

3 S. Zou et al. / Discrete Applied Matematics 156 (2008) H R (H) Fig. 1. A sextet-rotation grap R(H). Fig. 2. Four Clar covers, one containing two alternating exagons. M goes from a wite vertex to a black vertex, and improper oterwise. So a proper (resp. improper) sextet of M means a proper (resp. improper) M-alternating exagon. Te root perfect matcing of H is te unique perfect matcing witout proper sextets [13]. Given a perfect matcing M i of H, oter tan te root, te sextet rotation is a transform tat canges all proper sextets of M i into improper sextets, and leaves te oter edges uncanged. By tis, from M i anoter perfect matcing M j is obtained; we write tis as R(M i ) = M j. Te sextet-rotation digrap R(H) of H is constructed in te following manner: Its vertex set is te set of all perfect matcings of H, and tere is an arc from M i to M j if and only if R(M i ) = M j. An example (taken from te paper [13]) is presented in Fig. 1. A directed tree is an orientation of tree wit only one vertex of out-degree 0. Cen [1] sowed tat R(H) is a directed tree wit one root. A leaf of a directed tree is a vertex wose in-degree is 0. Te following is a well-known result, wic can be obtained by Teorem 3.6 in Ref. [4]. Lemma 2.1. A perfect matcing M of a exagonal system H corresponds to a non-leaf of R(H) if and only if eac proper M-alternating exagon (if suc exists) intersects some improper M-alternating exagon. Proof. Let M be a perfect matcing of H corresponding to a non-leaf of R(H). Ten H as anoter perfect matcing M suc tat R(M ) = M. Let S be te set of proper M -alternating exagons. So eac exagon in S is improper M-alternating. Furter, eac proper M-alternating exagon does not belong to S and intersects some exagon in S. Conversely, suppose tat eac proper M-alternating exagon (if suc exists) intersects some improper M-alternating exagon. Let S be te union of all improper M-alternating exagons. We ave tat S = by te above-mentioned result due to Gutman [5]. Taking te symmetric difference of M and te edge set of S, we get a perfect matcing M of H. Hence M = M and R(M ) = M; tat is, M is a non-leaf of R(H). A spanning subgrap of H is called a Clar cover if eac of its components is eiter a exagon or K 2. A exagon belonging to a Clar cover is often indicated by drawing a circle inside tis exagon; for example, see Fig. 2. Let C be te set of Clar covers witout alternating exagons in H. Let M be te set of perfect matcings corresponding to te non-leaves of R(H). Recall tat nl(h ) := M and cc(h ) := C. Teorem 2.2. Let H be a exagonal system wit a perfect matcing. Ten cc(h ) = nl(h ). (3) Proof. Define a mapping φ : M C as follows: For eac M M, let C M be te union of all improper M-alternating exagons of H and te oter edges of M. Ten C M is a Clar cover of H. By Lemma 2.1, eac proper M-alternating

4 1812 S. Zou et al. / Discrete Applied Matematics 156 (2008) exagon must intersect some improper M-alternating exagon. Terefore C M C and φ is a mapping. Next we sow tat φ is surjective. For any C C, place tree edges into eac exagon in C so tat tey form improper sextets, wereas te oter edges remain uncanged. By tis a perfect matcing M of H is obtained. Because C is a Clar cover witout alternating exagons, eac proper M-alternating exagon must intersect some exagon in C, wic is improper M-alternating. By Lemma 2.1, M belongs to M and φ(m) = C. Finally, for any perfect matcings M 1 and M 2 of M, suc tat φ(m 1 ) = φ(m 2 ) = C, H as te same improper M 1 - and M 2 -alternating exagons, and oter edges of M 1 and M 2 (not in alternating exagons) coincide. So M 1 = M 2 and φ is injective. Hence φ is a one-to-one correspondence from M to C and nl(h ) = cc(h ). 3. Sextet patterns Let G be a plane bipartite grap. From now on, for a subgrap H of G, G H always means G V(H), i.e. a subgrap obtained from G by deleting all vertices of H togeter wit teir incident edges. A subgrap H of G is said to be nice if G H as a perfect matcing. Obviously, a perfect matcing (if suc does exist) of a nice subgrap H can be extended to a perfect matcing of te entire grap. A face f of G is said to be resonant if its boundary is a nice cycle. A set S of disjoint interior faces of G is called a resonant pattern if G as a perfect matcing M suc tat all face-boundaries in S are simultaneously M-alternating cycles. Let K(G) and r(g) be te numbers of perfect matcings and resonant patterns of G, respectively. An edge of G is called allowed if it belongs to some perfect matcing of G; forbidden oterwise (see [12]). A connected bipartite grap wit a perfect matcing is said to be normal if it as no forbidden edges [19]. A generalized exagonal system (GHS) is a connected subgrap of a exagonal system. Te boundary of a GHS is te union of te boundaries of its infinite face and te non-exagonal finite faces (oles). For a exagonal system wit perfect matcings, a resonant pattern of H is called always a sextet pattern since it consists of exagons. Equivalently, a sextet pattern of H means a set of exagons of a Clar cover. Teorem 3.1 (Gutman et al. [7], Zang and Cen [21]). For a exagonal system H wit perfect matcings, r(h) K(H), and equality olds if and only if H contains no coronene (see Fig. 3) as its nice subgrap. By applying te concept of a g-cut line, we are able to give a simpler proof of Teorem 3.1. Definition 3.1 (Zang and Cen [22]). Let H be a GHS. A broken line L = P 3 is called a g-cut line of H (see Fig. 4) if: (1) and P 3 lie in te centers of two boundary edges of H; (2) if =,P 3, ten is te center of some exagonal face and P 3 = π/3; (3) te segments and P 3 are ortogonal to edge-directions; and (4) all te points in L lie in exagonal faces of H except for te degenerated case of = = P 3. In particular, if = or P 3, L is a cut line. Note wen some edge in H is not in any exagon te g-cut line passing troug it can degenerate to a point. Lemma 3.2 (Zang and Zang [27]). Let G be a connected plane bipartite grap wit perfect matcings. Assume tat te cycle C of G lies in te boundary of some face of G. If 2 1 V(C) independent edges of C are allowed, ten C is a nice cycle. Fig. 3. Coronene C 0.

5 S. Zou et al. / Discrete Applied Matematics 156 (2008) P 3 Fig. 4. Cut lines and a g-cut line P 3. e 1 P e 2 e 1 s e s+1 g 1 g 2 P 3 g t Fig. 5. Illustration of te proof of Lemma 3.3 (tick lines represent edges in M). Lemma 3.3. Let H be a GHS wit a perfect matcing. Te following statements old: (1) If H as a forbidden edge, ten tere exists a forbidden edge in te boundary of H [23]. (2) If a boundary edge of H is a forbidden edge, ten tere is a g-cut line L intersecting it, and all edges intersecting L are forbidden edges. Proof. Let e 1 be a forbidden edge of H.Ife 1 is a boundary edge of H, Statement (1) is trivial. Oterwise, let a exagon 1 of H contain e 1, and let e 2 be te edge of 1 opposite to e 1. If a exagon 2 of H (oter tan 1 ) contains e 2, let e 3 be te edge of 2 opposite to e 2. In tis way, we produce a series of parallel edges e 1,e 2,e 3,... (cf. Fig. 5). Let e s be te last forbidden edge in tis sequence; tat is, e 1,e 2,...,e s are forbidden and eiter tis sequence ends at e s or e s+1 is an allowed edge. If e s is a boundary edge, Statement (1) olds. Oterwise, suppose tat H as a exagon s ( = s 1 ) containing e s. Ten e s+1 is an edge of s opposite to e s. Hence e s+1 is in some perfect matcing M of H.By Lemma 3.2, one (say g) of two edges adjacent to e s in s is forbidden. Let L be a straigt segment from te center of te exagon s to te center P 3 of a boundary edge of H troug te center of edge g suc tat all te points of L lie in exagons of H. Let g 1 (=g), g 2,g 3,...,g t be te all edges intersecting L suc tat any consecutive g i and g i+1 are contained in a exagon i of H. Ten g t is a boundary edge and its center is P 3.Ift>1, bot edges adjacent to g 1 in 1 belong to M. Since g 1 is a forbidden edge, g 2 is also forbidden by Lemma 3.2. If t>2, bot edges adjacent to g 2 in 2 belong to M and g 3 is a forbidden edge. Continuing tis process we arrive in tat all te g i s are forbidden. Hence statement (1) olds. Now we coose a forbidden edge e 1 in te boundary of H in te above proof. Let be te center of e 1.Ife s is a boundary edge, is a required cut line. Oterwise, points and P 3 are te centers of a exagon s and an edge g t, respectively. Ten P 3 is a required g-cut line and statement (2) olds. Lemma 3.4 (Zang [24, Teorem 3.2.1]). LetGbea 2-connected plane bipartite grap wit perfect matcings. Ten r(g) K(G), and equality olds if and only if tere do not exist disjoint cycles R and C suc tat (a) R is a facial boundary lying in te interior of C and (b) C R is a nice subgrap of G.

6 1814 S. Zou et al. / Discrete Applied Matematics 156 (2008) L e v e Fig. 6. A GHS I[C]. A New Proof of Teorem 3.1. We only sow tat r(h) = K(H) if and only if H contains no coronene as its nice subgrap. Te necessity follows by Lemma 3.4. For sufficiency, suppose, to te contrary, tat r(h)<k(h). By Lemma 3.4, tere exist a exagon and a cycle C suc tat lies in te interior of C and H C as a perfect matcing. Let I[C] be te subgrap of H consisting of C togeter wit its interior. Ten I[C] is a GHS wit precisely one non-exagonal interior face, tat is ole (see Fig. 6), and its boundary C is a nice cycle. Denote by C te boundary of tis ole. We now sow tat I[C] is normal. Suppose tat I[C] as a forbidden edge. By Lemma 3.3, tere exists a g-cut line L = P 3 suc tat all edges intersecting L are forbidden. As C is a nice cycle of I[C], L can only be a broken line wit P 3 = π/3, and te two end-points of L lie on C. Since C is te boundary of coronene and P 3 = π/3, te end-points of L can only lie on te adjacent edges e and e of te same exagon. Let v be te vertex sared by e and e. Since bot e and e are forbidden in I[C], and v is of degree 2 and v cannot be matced to oter vertices of I[C]. Tis contradicts to te assumption tat I[C] as a perfect matcing. So I[C] is normal. Consequently, eac face of I[C] is resonant [23] and C is a nice cycle of I[C], wic implies tat te coronene spanned by and C is a nice subgrap of H, contradicting te condition of Teorem Caracterization of cs(h ) = cc(h ) A Clar cover witout alternating exagons is not necessarily a Clar structure. For example, te left-and side diagram in Fig. 7 is not a Clar structure of tribenzo[a,g,m]coronene, wereas te rigt-and side one is. On te oter and, bot diagrams are Clar covers witout alternating exagons. For any exagonal systems H, weavecs(h ) cc(h ) = nl(h ). For te exagonal system depicted in Fig. 1, all Clar covers witout alternating exagons of H are also Clar structures, as sown in Fig. 2; ence, in tis case, cs = cc. It is natural to pose te question wen bot quantities are equal. We first give a sufficient condition for tis. Lemma 4.1 (Zang and Zang [27]). Let G be a plane elementary bipartite grap wit a perfect matcing M and let C be an M-alternating cycle. Ten tere exists an M-resonant face in I[C], were I[C] denotes te subgrap of H consisting of C togeter wit its interior. Teorem 4.2. If a exagonal system H as a perfect matcing and contains no coronene as its nice subgrap, ten cs = cc. Proof. Suppose tat cs < cc. Ten tere exists a Clar cover C witout alternating exagons in H, wic is not a Clar structure of H. Let M be a perfect matcing of H corresponding to C, suc tat all exagons in C are proper M-alternating. Since C is not a Clar structure, tere exists anoter perfect matcing M in H, different from M, suc tat all exagons in C are proper M -alternating. Ten tere is an M and M-alternating cycle C in M M = (symmetric difference). We claim tat te interior of C contains at least one exagon of C. Oterwise, by Lemma 4.1 I[C] would contain an M-alternating exagon wic would be disjoint from any exagon in C. Tis contradicts to C being a Clar cover of H witout alternating exagons. As C and are disjoint M-alternating cycles, and lies in te interior of C, by Teorem 3.1 and Lemma 3.4, H as a coronene as its nice subgrap, a contradiction. Tus cs = cc.

7 S. Zou et al. / Discrete Applied Matematics 156 (2008) Fig. 7. Two Clar covers of tribenzo[a,g,m]coronene. Fig. 8. Te parallelogram L m,n. Fig. 9. Two exagonal systems wit cs = cc. Corollary 4.3. For te parallelogram L m,n (see Fig. 8), cs = cc. Proof. Draw a cut line L in eac row of L m,n suc tat L intersects only vertical edges. Let I denote te set of edges of L m,n intersecting L. L m,n I (te removal of all edges in I from L(m, n)) possesses exactly two components and te difference between te numbers of wite and black vertices in eac component is one. Ten eac perfect matcing M of L m,n contains exactly one edge in I. Similarly, draw a cut line L in te middle row of coronene (C 0 ) and denote te set of edges intersecting L in coronene by I. Since te difference between te numbers of wite and black vertices in eac component of C 0 I is two, every perfect matcing M of C 0 contains exactly two edges of I. Tus coronene is not a nice subgrap of te parallelogram L m,n. By Teorem 4.2, cs = cc. Corollary 4.4. For te exagonal systems sown in Fig. 9, cs = cc. Proof. By a similar argument as used in Corollary 4.3, we can sow tat te exagonal systems in Fig. 9 contain no coronene as teir nice subgrap. By Teorem 4.2, we ten ave cs = cc. Te converse of Teorem 4.2 does not old. For example, as Fig. 10 sows, all Clar covers witout alternating exagons in coronene are identical to teir Clar structures. Hence cs(c 0 )=cc(c 0 ). So we can obtain te Clar polynomial of coronene by enumerating Clar covers witout alternating exagons as follows: ρ(x, C 0 ) = 2x 3 + 3x 2 + 2x wic, of course, agrees wit te earlier result of [14]. For anoter exagonal system H wit cs = cc in Fig. 11, ina similar manner we get ρ(x, H ) = 3x 4 + 6x 3 + 3x 2.

8 1816 S. Zou et al. / Discrete Applied Matematics 156 (2008) Fig. 10. All Clar structures of coronene. Fig. 11. All Clar structures of a exagonal system. We now give a necessary and sufficient criterion for exagonal systems wit cs = cc. Teorem 4.5. Let H be a exagonal system wit perfect matcings. Ten cs = cc if and only if for eac Clar cover C witout alternating exagons in H, H C s does not ave a cycle C intersecting a exagon along a pat of odd lengt suc tat C is a nice subgrap of H C s, were C s denotes te set of exagons in C. Proof. We prove te contrapositive statement of te teorem. Tat is, cs < cc if and only if for some Clar cover C witout alternating exagons in H, H C s as a cycle C intersecting a exagon along a pat of odd lengt, suc tat C is a nice subgrap of H C s. Sufficiency: For a Clar cover C of H witout alternating exagons, suppose tat te intersection of a exagon and cycle C in H C s is a pat of odd lengt and C is a nice subgrap of H C s. Ten and C are in one of te modes L 1, L 3, L 5 (cf. Fig. 15). Let P := C. Since P is a pat of odd lengt, C P is a pat of odd lengt and P is a pat of odd lengt or empty. Taking te perfect matcings of tese tree pats, we ave tat teir union forms a perfect matcing of C. Since C is a nice subgrap of H C s, te perfect matcing of C can be extended to a perfect matcing M of H C s.asis an M-alternating exagon in H C s, H (C s ) as a perfect matcing. Tus te exagons in C s and a perfect matcing of H (C s ) compose a Clar cover C of H. As C s C s, we conclude tat C is not a Clar structure of H. Hence cs < cc. Necessity: Suppose cs < cc. Ten tere must exist a Clar cover C witout alternating exagons in H, butcis not a Clar structure of H. So tere is anoter perfect matcing M in H suc tat all exagons in C are proper M -alternating,

9 S. Zou et al. / Discrete Applied Matematics 156 (2008) v 1 v 6 v 2 v 3 v 5 v 4 v 1 v 6 v 5 v 2 v3 v 4 v 1 v 2 v 6 v 3 v v5 4 M M Fig. 12. Illustration to te proof of Teorem 4.5. l l 2 m 1 n l 1 P l H 1 H 2 H 3 H 4 m Fig. 13. Examples of exagonal systems (l, m, n 2) wit cs = cc. Fig. 14. All Clar structures of te exagonal system H 1. and tere exists at least one M -alternating exagon in H C s. Let M be a perfect matcing of H corresponding to C, suc tat all exagons in C are proper M-alternating. Since is M -alternating but not M-alternating, tere is an M and M -alternating cycle C in M M intersecting. Ten C H C s and C =. Tere exists a pat P in C wic is internally disjoint from exagon, and te two end-vertices (say, v 1 and v 2 )ofp lie on (see Fig. 12). Because bot C and are M -alternating cycles, P is an M -alternating pat, bot end-edges of wic are not in M. Hence te restriction of M on P is its perfect matcing and P is a nice subgrap of H. Since P is a pat of odd lengt, its end vertices v 1 and v 2 are of distinct colors. Hence is divided into two pats of odd lengt by te pair of vertices v 1 and v 2, and P can be expressed as te union of a cycle C and, suc tat C is a pat of odd lengt. Corollary 4.6. For te exagonal systems sown in Fig. 13, cs = cc. Proof. Draw a cut line L of H 1 as sown in Fig. 13. For eac perfect matcing M of H 1, tere is exactly one edge in M, intersecting L. Tat is, for eac Clar cover C witout alternating exagons in H 1, tere is exactly one exagon wic intersects L belonging to C. Delete te exagon and bot end-vertices of all edges wic lie in all perfect matcings of H 1 from H 1. If exagon 1 belongs to C s, ten te resulting grap is isomorpic to grap H 1 sown in Fig. 14. If one of te exagons 2,...,l belongs to C s, te resulting grap is isomorpic to coronene. Eac Clar

10 1818 S. Zou et al. / Discrete Applied Matematics 156 (2008) cover witout alternating exagons in H 1 (or in coronene) (see Figs. 10 and 14) togeter wit te exagon 1 (or one of te exagons 2,...,l) and oter K 2 components induce a Clar cover of H 1 witout alternating exagons. Clearly tese are all te Clar covers witout alternating exagons in H 1. Hence for eac Clar cover C of H 1 witout alternating exagons, H 1 C s as no cycles C intersecting a exagon at a pat of odd lengt. By Teorem 4.5, cs(h 1 ) = cc(h 1 ). By te same arguments, we can deduce tat equation cs = cc olds for te oter tree exagonal systems in Fig. 13. By Corollary 4.6, we can obtain te Clar polynomials of tese exagonal systems by constructing Clar covers witout alternating exagons. Suc a computing is exemplified for H 1. Consider te Clar covers C of H 1 witout alternating exagons, and assume tat te exagon i belongs to C s.ifi 2, ten ρ 1 (x, H 1 ) :=(l 1) x ρ(x, C o ). If te exagon 1 belongs to C s, ten ρ 2 (x, H 1 )=x ρ(x, H 1 )=x(x 3 +3x 2 +x), were H 1 is te exagonal system sown in Fig. 14. Adding te two above polynomials we obtain te Clar polynomial of H 1 : ρ(x, H 1 ) = ρ 1 (x, H 1 ) + ρ 2 (x, H 1 ) = (2l 1)x 4 + 3lx 3 + (2l 1)x 2. In a similar manner, we obtain also te following Clar polynomials: ρ(x, H 2 ) = (2ml 3m 2l + 3)x 5 + (3ml 3m 3l + 5)x 4 + (2ml 3m 2l + 6)x 3 + 2x 2, ρ(x, H 3 ) = (5l + 1)x 4 + (l + 2)x 2, ρ(x, H 4 ) = 2mnlx 6 + l(m + n)x 5 +[(2 + l)mn + m + n + l + 1]x 4 + x Construction of exagonal systems wit cs < cc Tere are many exagonal systems wit cs < cc. We now give a construction approac, based on a series of normal additions of exagons, starting from coronene suc tat in eac step te coronene is a nice subgrap of te exagonal system. We recall te concept of normal additions. Under a normal addition [6] is understood an addition of a new exagon to a exagonal system, suc tat te added exagon acquires te modes L 1, L 3,orL 5 (see Fig. 15). In fact a normal addition of a exagon is to add a pat of lengt 1, 3 or 5 to a exagonal system H suc tat bot end vertices identify vertices of distinct colors in H, it is internally disjoint H and te resultant is a larger exagonal system. Suc pats of odd lengt are called ears. For a normal exagonal system te following construction was originally conjectured by Cyvin and Gutman [2], and eventually rigorously proved by He and He [10]. Teorem 5.1 (He and He [10]). Any normal exagonal system wit + 1 exagons can be generated from a normal exagonal system wit exagons by a normal addition of one exagon. We call a construction specified in Teorem 5.1 a normal construction. All exagonal systems in Figs. 11 and 13 ave cs = cc, and can be obtained by normal constructions starting from teir nice subgrap coronene. In order to obtain a exagonal system wit cs < cc by normal construction, some furter conditions must be needed. By C 0 (see Fig. 16) we denote te exagonal system obtained by attacing to coronene C 0 a new exagon in mode L 1. In tis section we also assume tat all cycles considered are oriented clockwise, and, witout loss of L 1 L 3 L 5 Fig. 15. Tree modes of normal additions.

11 S. Zou et al. / Discrete Applied Matematics 156 (2008) Fig. 16. H 0 = C 0. m 1 2 n m Fig. 17. Hexagonal system C 2m 1,n. generality, te vertices of C 0 are colored so tat starts at a wite vertex of C 0 along te boundary (C 0 ) of C 0. Suppose tat H as a normal construction (H 0,H 1,...,H r =H), associated wit te ear sequence (P 0,,...,P r 1 ) and exagon sequence (S 0,S 1,...,S r 1 ). Eac ear P i is added to H i to get H i+1 so tat only two end-vertices of P i lie on te boundary H i of H i, adding ear P i to H i is equivalent to a normal addition of exagons S i, and P i starts and ends at vertices of H i in te sense of te clockwise orientation of H i+1,i= 0, 1,...,r 1. Teorem 5.2. Let H be a exagonal system wit a perfect matcing. If H as a normal construction (H 0 = C 0, H 1,...,H r = H), associated wit te ear sequence (P 0,,...,P r 1 ) and exagon sequence (S 0,S 1,...,S r 1 ), suc tat if te two end-vertices of ear P i lie on H 0, and P i and start at vertices of different colors, te two end-vertices of te ear P i+1 must lie on S i, and P i+1 and start at vertices of te same color, ten cs < cc. Proof. Construct a perfect matcing M of H suc tat H 0 is an improper M-alternating cycle, te central exagon of coronene C 0 is improper M-alternating, and eac ear P i is an M-alternating pat, i = 1, 2,...,r 1. Te union of all improper M-alternating exagons and te oter edges from M form a Clar cover C of H. If tere is no oter M-alternating exagon in H C s left, C is a Clar cover witout alternating exagons in H. Oterwise, tere exist oter M-alternating exagons in H C s. Tese exagons must be proper M-alternating. Hence tese are mutually disjoint and also disjoint from te exagons in C s. Ten te proper M-alternating exagons togeter wit te oter edges of M in H C s form a Clar cover C. Hence C := C s C is a Clar cover witout M-alternating exagons in H. Ifan ear wose end-vertices lie on H 0 starts at a vertex of te same color as, ten te added exagon intersects H 0 and cannot be M-alternating. If it as at most one end-vertex lying on H 0 and te added exagon intersects H 0, ten tat te exagon is not M-alternating. Tus it does not belong to C. So assume tat tere exists at least one ear, say P i, wose end-vertices lie on H 0, and P i starts at a vertex of color different tan. Altoug te exagon S i is proper M-alternating, S i+1 is improper M-alternating since te end-vertices of te ear P i+1 lie on S i, and P i+1 starts wit te same color as. By te coice of C, S i+1 belongs to C. Since S i and S i+1 ave an edge in common, S i cannot belong to C. In eac case, C s H 0 =. Te exagon is tus an M C-alternating exagon in H C s. Tat is, te set of exagons in C is not maximal. Hence C is not a Clar structure of H and cs < cc (cf. Teorem 4.5). In Fig. 7, te two exagons added to H 0 start wit te same color as. By using Teorem 5.2, we ave tat cs < cc olds for tribenzo[a,g,m]coronene. Trivially, for H 0 = C 0, cs < cc. Corollary 5.3. For te exagonal system C 2m 1,n (m>1) (see Fig. 17), if n m + 1 2, ten cs < cc.

12 1820 S. Zou et al. / Discrete Applied Matematics 156 (2008) P 4 P 3 P 3 P 5 H 0 H 2 H 3 H 5 P i+1 P i+k P i+2 P i k 1 2 k 1 2 k P k P k+2 P k+1 P l+1 P l+k P l+2 P l+1 H i H i+2 Fig. 18. Te main steps of normal construction of C 2m 1,n for k 3. H P 7 m P 4 P 3 P 5 P 6 P m n H 0 H 1 H 6 H i H Fig. 19. Te main steps of normal construction of C 2m 1,n for k = 2. P 3 P 4 Fig. 20. Hexagonal system H. Proof. Let n m + 1 = k. Ten bot te top and te bottom of C 2m 1,n ave k (k 2) exagons. We can give a normal construction of C 2m 1,n from H 0 wic satisfies te conditions of Teorem 5.2. Figs. 18 and 19 sketc suc a construction, pertaining to te two cases: k 3 and k = 2. Te details are omitted. Corollary 5.4. Let H be a normal exagonal system wit a perfect matcing. If H contains H (see Fig. 20) as its nice subgrap, ten cs < cc. Proof. Since H is a nice subgrap of H, H as a normal construction starting from H (cf. [27]). As for H, it as a normal construction starting from H 0 = C 0 as follows. First add ears and of mode L 3 to H 0, so tat starts

13 S. Zou et al. / Discrete Applied Matematics 156 (2008) at a wite vertex of C 0 and at a wite vertex of. Ten add te ear P 3 of mode L 1 so tat P 3 starts at a wite vertex of C 0. Finally, add te ear P 4 of mode L 3 so tat P 4 starts at a wite vertex of C 0. Hence H as a normal construction starting from H 0 = C 0. Since te ears wit two end-vertices lying on H 0 can only start wit te same color as, by Teorem 5.2, cs < cc. Toug we ave given some examples constructing exagonal systems wit cs < cc by using Teorem 5.2, we are not sure weter tis metod can be used to construct all exagonal systems wit cs < cc. Acknowledgement Te autors are grateful to te referees for teir careful reading and many valuable suggestions. References [1] Z. Cen, Directed tree structure of te set of Kekulé patterns of polyex graps, Cem. Pys. Lett. 115 (1985) [2] S.J. Cyvin, I. Gutman, Kekulé Structures in Benzenoid Hydrocarbons, Springer, Berlin, [3] S. El-Basil, Combinatorial Clar sextet teory, Teor. Cim. Acta 70 (1986) [4] X. Guo, F. Zang, An efficient algoritm for generating all Kekulé patterns of a generalized benzenoid system, J. Mat. Cem. 12 (1993) [5] I. Gutman, Covering exagonal systems wit exagons, in: Proceedings of te 4t Yougoslav seminar on Grap teory, Novi Sad, 1983, pp [6] I. Gutman, S.J. Cyvin, Introduction to te Teory of Benzneoid Hydrocarbons, Springer, Berlin, [7] I. Gutman, H. Hosoya, T. Yamaguci, A. Motoyama, N. Kuboi, Topological properties of benzenoid systems. III. Recursion metod for te sextet polynomial, Bull. Soc. Cim. Beograd 42 (1977) [8] I. Gutman, A.V. Teodorović, N. Kolaković, Algebraic studies of Kekulé structures, A semilattice based on te sextet rotation concept, Z. Naturforsc 44a (1989) [9] I. Gutman, A.V. Teodorović, N. Kolaković, An application of corals, J. Serb. Cem. Soc. 55 (1990) [10] W.C. He, W.J. He, Some topological properties of normal benzenoids and coronoids, MATCH Commun. Mat. Comput. Cem. 25 (1990) [11] W.C. Herndon, H. Hosoya, Parameterized valence bond calculations for benzenoid ydrocarbons using Clar structure, Tetraedron 40 (1984) [12] L. Lovász, M.D. Plummer, Matcing Teory, Nort-Holland, Amsterdam, [13] N. Okami, A. Motoyama, H. Hosoya, I. Gutman, Grap-teoretical analysis of te Clar s aromatic sextet, Tetraedron 37 (1981) [14] M. Randić, S. El-Basil, Grap teoretical analysis of large benzenoid ydrocarbons, J. Mol. Struct. (Teocem) 303 (1994) [15] M. Randić, S. El-Basil, S. Nikolić, N. Trinajstić, Clar polynomial of large benzenoid systems, J. Cem. Inf. Comput. Sci. 38 (1998) [16] H. Sacs, Perfect matcings in exagonal systems, Combinatorica 4 (1984) [17] L.J. Scaad, B.A. Hess, Dewar resonance energy, Cem. Rev. 101 (2001) [18] W.C. Siu, P.C.B. Lam, H. Zang, Clar and sextet polynomials of buckminsterfullerene, J. Mol. Struct. (Teocem) 622 (2003) [19] W.C. Siu, P.C.B. Lam, F. Zang, H. Zang, Normal components, Kekulé patterns and Clar patterns in plane bipartite graps, J. Mat. Cem. 31 (2002) [20] R. Swinborne-Seldrake, W.C. Herndon, I. Gutman, Kekulé structures and resonance energies of benzenoid ydrocarbons, Tetraedron Lett. 16 (1975) [21] F. Zang, R. Cen, A teorem concerning polyex graps, MATCH Commun. Mat. Comput. Cem. 19 (1986) [22] F. Zang, R. Cen, Wen eac exagon of a exagonal system covers it, Discr. Appl. Mat. 30 (1991) [23] F. Zang, M. Zeng, Generalized exagonal systems wit eac exagon being resonant, Discr. Appl. Mat. 36 (1992) [24] H. Zang, Perfect matcings in plane elementary bipartite grap, P.D. Tesis, Sicuan University, [25] H. Zang, P.C.B. Lam, W.C. Siu, Cell rotation graps of strongly connected orientations of plane graps wit an application, Discr. Appl. Mat. 130 (2003) [26] H. Zang, F. Zang, Te Clar covering polynomial of exagonal systems I, Discr. Appl. Mat. 69 (1996) [27] H. Zang, F. Zang, Plane elementary bipartite graps, Discr. Appl. Mat. 105 (2000)