All the circuits of a catacondensed benzenoid system are conjugated

Khaled Salem

The British University in Egypt, El Sherouk City, Misr-Ismalia Desert Road, Postal No. 11837, P.O. Box 43, Egypt

Received 7 April 2006; accepted 31 May 2006

Available online 17 July 2006

Abstract

It is shown that all the circuits of a catacondensed benzenoid system are conjugated.

q 2006 Elsevier B.V. All rights reserved.

Keywords: Benzenoid system; Catacondensed; Conjugated circuit; Kekule structure

1. Introduction

Benzenoid hydrocarbons can be represented by graphs,

known as benzenoid systems, and so they lend themselves to

graph-theoretic analysis. There are many mathematical aspects

of benzenoid systems that are also significant in the chemistry

of benzenoid hydrocarbons [1,2]. For example, Randic used

conjugated circuits (also called alternating cycles in mathe-

matical literature) of a benzenoid system to estimate the

resonance energy of the corresponding benzenoid hydrocarbon

[3,4]. A class of benzenoid systems is called catacondensed.

Here, we show that all the circuits of a catacondensed

benzenoid system are conjugated. We think that this result

will have significant implications in the conjugated-circuits

model of Randic. In Section 2, we prove the result. In the

remainder of this section, we give some definitions, but

throughout this note, we assume some knowledge of graph

theory [5].

Let C be a circuit on the hexagonal lattice. Then the vertices

and the edges lying on C and in the interior of C form a

benzenoid system, B say [6]. We call C the defining circuit of

the benzenoid system B. The vertices of B lying on C are called

external vertices and the vertices of B lying in the interior of C,

if any, are called internal vertices. A benzenoid system is

called catacondensed if it has no internal vertices, otherwise it

is called pericondensed [7,8]. Benzene, a single hexagon, is the

trivial catacondensed benzenoid system.

In a catacondensed benzenoid system other than benzene,

one can distinguish among four types of hexagons [9]:

0166-1280/$ - see front matter q 2006 Elsevier B.V. All rights reserved.

doi:10.1016/j.theochem.2006.05.051

E-mail address: ksalem@bue.edu.eg.

(1) a hexagon adjacent to exactly one hexagon, called terminal;

(2) a hexagon adjacent to two hexagons, so that the edges that it

shares with its neighbors are disjoint and parallel, called

linearly annelated; (3) a hexagon adjacent to two hexagons,

so that the edges that it shares with its neighbors are disjoint but

not parallel, called angularly annelated; and (4) a hexagon

adjacent to three hexagons, so that the edges that it shares with

its neighbors are pair-wise disjoint, called branched.

A matching in a graph is an edge subset that are pair-wise

disjoint. A matching M is perfect if each vertex is incident with

some edge in M. A perfect matching is also called a Kekule

structure. A graph that has a perfect matching is called

Kekulean. A circuit C of a graph B is conjugated if there

exists a perfect matching M of B such that the edges of C are

alternately in M and its complement. A conjugated circuit C

may also be called M-conjugated to specify a corresponding

perfect matching M. An edge of a Kekulean graph is fixed if it

belongs to all or none of the perfect matchings of the graph. It

can be seen that an edge is not fixed if and only if it belongs to a

conjugated circuit. Gutman and Cyvin [10] showed that a

conjugated circuit in a benzenoid system has length 4mC2

for some positive integer m.

A hexagon R of a benzenoid system B is resonant if the

subgraph of B obtained by deleting from B the vertices of R has

a perfect matching or is empty [11,12]. Zhang and Chen [13]

showed that in a Kekulean benzenoid system B, each hexagon

of B is resonant if and only if B has no fixed edges.

2. The theorem and its proof

We introduce some notation. Let B be a catacondensed

benzenoid system other than benzene and R be a hexagon of B.

The subgraph of B obtained by deleting from B the vertices and

Journal of Molecular Structure: THEOCHEM 767 (2006) 189–191

www.elsevier.com/locate/theochem

R e

Fig. 2. The catacondensed benzenoid system B, where R is terminal.

R e

Fig. 3. The catacondensed benzenoid system B, where R is linearly annelated.

e2

K. Salem / Journal of Molecular Structure: THEOCHEM 767 (2006) 189–191190

the edges belonging to R, but not to any other hexagon of B is

denoted by BK{R}.

Lemma 1. Let B be a catacondensed benzenoid system other

than benzene. Let R be a hexagon of B. If R is terminal, then

BK{R} is a catacondensed benzenoid system, if R is linearly

annelated or angularly annelated, then BK{R} has two

connected components each of which is a catacondensed

benzenoid system, and if R is branched, then BK{R} has

three connected components each of which is a catacondensed

benzenoid system.

A less rigorous statement of Lemma 1, that is also less

detailed, was given by Gutman et al. [9]. Lemma 1 can be

verified by inspecting the structure of the catacondensed

benzenoid system B for all the modes of the hexagon R.

Lemma 2. Each hexagon of a catacondensed benzenoid system

is resonant.

In 1975, Hosoya and Yamaguchi [11] noted, without proof,

that Lemma 2 is true for almost all the cases. In fact, they gave

the central hexagon in perylene, a pericondensed benzenoid

system, as a counterexample. In 1977, Gutman et al. [9] stated

Lemma 2, but again without proof. Finally, in 1983, Gutman

[12] published a proof.

Theorem 1. A circuit C of a catacondensed benzenoid system B

is conjugated.

Proof: We use induction on the number of hexagons of B

enclosed by C, r say. Initial step: the result is true for rZ1 by

Lemma 2. Inductive step: assume that the result is true for rZk.

We show that it is true for rZkC1, where kR1.

Let BC be the benzenoid system defined by C. It is clear that

the hexagons of B enclosed by C are the hexagons of BC and

that BC is catacondensed. Let R be a terminal hexagon of BC.

Then BCK{R} is a catacondensed benzenoid system by

Lemma 1. The catacondensed benzenoid system BC is shown

in Fig. 1.

Let C 0 be the defining circuit of BCK{R}. It is clear that C 0

is a circuit of B and it encloses k hexagons of B. Thus, by the

inductive assumption, it is a conjugated circuit of B. Let M be a

perfect matching of B such that the edges of C 0 are alternately

in M and its complement. Let e be the edge shared between R

and the hexagon of BC adjacent to it. See Fig. 1. We can assume

that the edge e does not belong to M. We consider all the

possible modes of R as a hexagon of B.

Case: R is terminal in B. The catacondensed benzenoid

system B is shown in Fig. 2. It is easy to see from the figure that

C is M-conjugated. Case: R is linearly annelated in B. The

R e BC -{R}

Fig. 1. The catacondensed benzenoid system BC.

catacondensed benzenoid B is shown in Fig. 3. It is easy to see

from the figure that C is M-conjugated.

Case: R is angularly annelated in B. The catacondensed

benzenoid system B is shown in Fig. 4. The edge e2 shown in

Fig. 4 belongs to M. Subcase: The edge e1 shown in Fig. 4

belongs to M. Then C is M-conjugated. Subcase: The edge e1

shown in Fig. 4 does not belong to M. Consider the catacon-

densed benzenoid system B 0 shown in the figure (recall Lemma

1). Let M 0 be a perfect matching of the defining circuit of B 0

such that e12M 0. It is obvious that M 0 is a perfect matching of

B 0. It is also easy to see that (MKE(B 0))gM 0 is a perfect

matching of B such that the edges of C are alternately in it and

its complement. Thus, C is a conjugated circuit of B.

Case: R is branched in B. The catacondensed benzenoid

system B is shown in Fig. 5. Let e1 and e2 be the edges shown in

the figure. Subcase: Both e1 and e2 belong to M. Then C is

M-conjugated. Subcase: Exactly one of e1 and e2 belong to M.

It can be shown that C is conjugated in B. The argument is

analogous to an angularly annelated subcase. Subcase: Neither

e1 nor e2 belong to M. First, we show that the edge e 0 shown in

Fig. 5 does not belong to M.

Assume that e 02M. Consider the catacondensed benzenoid

system B 0 shown in the figure (recall Lemma 1). Let v be the

vertex of B 0 that is incident to e 0. Note that V(B 0)K{v} is the

vertex set of the subgraph of B 0 induced by MhE(B 0). Since, B 0

has an even number of vertices, the cardinality of V(B 0)K{v} is

odd. Since, M is a matching, the cardinality of the vertex set of

R

B'

e

e1

e2

R

B'

e

e1

Or

Fig. 4. The catacondensed benzenoid system B, where R is angularly annelated.

R

B'

B''

e

e2

e1

e'

Fig. 5. The catacondensed benzenoid system B, where R is branched.

K. Salem / Journal of Molecular Structure: THEOCHEM 767 (2006) 189–191 191

the subgraph of B 0 induced by MhE(B 0) is even, a contradic-

tion. Hence, e 0;M.

Let M 0 and M 00 be perfect matchings of the defining circuits

of the catacondensed benzenoid systems B 0 and B 00 shown in

Fig. 5 such that e12M 0 and e22M 00. It is not difficult to see that

ðMKðEðB0ÞgEðB00ÞÞÞg ðM 0gM 00Þ is a perfect matching of B

and that the edges of C are alternately in this perfect matching

and its complement. Thus, C is a conjugated circuit of B. ,

Acknowledgements

The author thanks Professor Ivan Gutman (Faculty of

Science, University of Kragujevac, Serbia and Montenegro)

for confirming the originality of the result. The author also

thanks Professor Hernan Abeledo (Engineering Management

and Systems Engineering, The George Washington University,

USA) for drawing his attention to research on benzenoid

systems and for many fruitful discussions.

References

[1] I. Gutman, S.J. Cyvin, Introduction to the Theory of Benzenoid Hydro-

carbons, Springer, Berlin, 1989.

[2] M. Randic, Chem. Rev. 103 (2003) 3449.

[3] M. Randic, Chem. Phys. Lett. 38 (1976) 68.

[4] M. Randic, Tetrahedron 33 (1977) 1905.

[5] F. Harary, Graph Theory, Addison-Wesley, Reading, MA, 1969.

[6] I. Gutman, Bull. Soc. Chim. Beograd 47 (1982) 453.

[7] A.T. Balaban, F. Harary, Tetrahedron 24 (1968) 2505.

[8] I. Gutman, Croat. Chem. Acta 46 (1974) 209.

[9] I. Gutman, H. Hosoya, T. Yamaguchi, A. Motoyama, N. Kuboi, Bull. Soc.

Chim. Beograd 42 (1977) 503.

[10] I. Gutman, S.J. Cyvin, J. Mol. Struct. (Theochem) 184 (1989) 159.

[11] H. Hosoya, T. Yamaguchi, Tetrahedron Lett. 52 (1975) 4659.

[12] I. Gutman, Wiss. Z. Thechn. Hochsch. Ilmenau, 29 (1983) 57.

[13] F. Zhang, R. Chen, Discrete Appl. Math. 30 (1991) 63.