The Clar Structure of Fullerenes

The Clar Structure of Fullerenes Liz Hartung Massachusetts College of Liberal Arts June 12, 2013 Liz Hartung (Massachusetts College of Liberal Arts) The Clar Structure of Fullerenes June 12, 2013 1 / 25

Introduction A Fullerene is a trivalent plane graph Γ = (V, E, F ) with only pentagonal and hexagonal faces. Fullerenes are designed to model carbon molecules. The vertices represent carbon atoms and the edges represent the chemical bonds between them. Liz Hartung (Massachusetts College of Liberal Arts) The Clar Structure of Fullerenes June 12, 2013 2 / 25

Kekulé Structure Carbon atoms form four covalent bonds. Three of these are "strong" bonds, and one is "weak". In the graphical representation, we draw the three strong bonds. The fourth bond tends to occur as a double bond. Over a fullerene, the edges of a perfect matching correspond to double bonds, called a Kekulé structure. Liz Hartung (Massachusetts College of Liberal Arts) The Clar Structure of Fullerenes June 12, 2013 3 / 25

Fries Number A face in Γ can have 0,1,2, or 3 of its bounding edges in K. The faces that have 3 of their bounding edges in K are called the benzene faces of K. The Fries number of Γ is the maximum number of benzene faces over all Kekule structures of Γ Liz Hartung (Massachusetts College of Liberal Arts) The Clar Structure of Fullerenes June 12, 2013 4 / 25

Clar Number The Clar number of Γ is the largest independent set of benzene faces over all Kekule structures for Γ. Liz Hartung (Massachusetts College of Liberal Arts) The Clar Structure of Fullerenes June 12, 2013 5 / 25

Face 3-colorings We can find lower bounds for the Fries and Clar number of a fullerene based on a partial face 3-coloring. In a face 3-coloring of a hexagonal patch, we can choose one color class (blue) to be the set of void faces in the Kekulé structure. Let all of the edges not bounding these blue faces be in the Kekulé structure. Over the patch, all faces that are not void are benzene faces. We can choose one of the remaining color classes to be the independent set of benzene faces contributing to the Clar number. Liz Hartung (Massachusetts College of Liberal Arts) The Clar Structure of Fullerenes June 12, 2013 6 / 25

Overview For a fullerene Γ = (V, E, F) the Clar number is bounded above by V 6 2. For small fullerenes, the Clar and Fries numbers have been found through computer searches. We introduce Clar chains, a decomposition of fullerenes. We use Clar chains to construct an improper face 3-coloring of the fullerene. This allows us to compute the Clar number directly for certain classes of fullerenes. Liz Hartung (Massachusetts College of Liberal Arts) The Clar Structure of Fullerenes June 12, 2013 7 / 25

Clar Structures A Clar Structure (C, A) for a fullerene Γ is a set of hexagons, C, and edges, A, such that every vertex is incident with exactly one element of C A and at most two edges of A lie on any face of Γ. Given a Clar structure (C, A), we can choose three alternating edges on each face in C. Together with the edges of A, these edges form a Kekulé structure K associated with the Clar structure (C, A). Conversely, given a Kekulé structure K, we can form a Clar structure (C, A): take a maximal independent set of benzene faces to be the set C, and let the remaining edges of K form the set A. Liz Hartung (Massachusetts College of Liberal Arts) The Clar Structure of Fullerenes June 12, 2013 8 / 25

Clar Structures and the Clar Number The Clar number of a fullerene is given by a Clar structure (C, A) with a maximum number of faces in C. Lemma Let Γ = (V, E, F ) be a fullerene with a Clar structure (C, A). Then C = V 6 A 3. Every face in C contains six vertices and every edge in A contains two vertices, so 6 C + 2 A = V. We want to maximize the number of independent benzene faces, so our goal will be to find a Clar structure (C, A) that minimizes A. Liz Hartung (Massachusetts College of Liberal Arts) The Clar Structure of Fullerenes June 12, 2013 9 / 25

Constructing an Improper Face 3-coloring We will construct an improper face-3 coloring that retains the Clar structure (C, A). For a face f of Γ, we say that an edge a A exits f if a shares exactly one vertex with f and that a lies on f if both vertices of a are incident with f. For a face f of Γ, any face of C adjacent to f is incident with two adjacent vertices on f, as is any edge from A that lies on f. f f f Thus an odd number of edges of A exit a pentagon, an even number of edges of A (possibly zero) exit a hexagon. Liz Hartung (Massachusetts College of Liberal Arts) The Clar Structure of Fullerenes June 12, 2013 10 / 25

Clar Chains For each face f of Γ, we construct a coupling of the edges of A exiting f : Choose edges exiting from adjacent vertices to form a couple if such a pair exists. Continue coupling exit edges from adjacent vertices until either: all edges are coupled; only a pair of edges exiting from opposite vertices of a hexagon remains and these can be coupled; only a single exit edge of a pentagon remains. For each pentagon, we call the uncoupled exit edge the initial edge for that pentagon. Liz Hartung (Massachusetts College of Liberal Arts) The Clar Structure of Fullerenes June 12, 2013 11 / 25

Clar Chains Each edge in A that is not an initial edge from a pentagon has two couplings, one over each face that it exits. The twelve initial edges have at most one coupling. Given a coupling for a fullerene Γ, define a Clar chain in Γ to be an alternating sequence f 0, a 1, f 1, a 2,..., a k, f k of faces f i of Γ and edges a i in A such that a i and a i+1 are coupled edges exiting f i. If f 0 = f k, we say the chain is closed. Otherwise, it is an open chain, where f 0 and f k are the initial edges of pentagons. Liz Hartung (Massachusetts College of Liberal Arts) The Clar Structure of Fullerenes June 12, 2013 12 / 25

Clar Chains The assigned coupling of exit edges over each face determines six open chains between pairs of pentagons and ensures that each edge in A is included in exactly one chain. The chains may share faces but do not cross one another. Lemma Let Γ be a fullerene with a Clar structure (C, A) and a coupling assignment. There are exactly six open Clar chains connecting pairs of pentagons. There may additionally be closed Clar chains. Liz Hartung (Massachusetts College of Liberal Arts) The Clar Structure of Fullerenes June 12, 2013 13 / 25

Expansion and Face 3-coloring We define the expansion E(C, A) to be the graph obtained by widening the edges in A into quadrilateral faces. Each vertex incident with an edge in A becomes an edge, and each edge in A splits lengthwise into two edges. All faces in the expansion are of even degree. Theorem (Saaty and Kainen ) A 3-regular plane graph is face 3-colorable if and only if each face has even degree. Liz Hartung (Massachusetts College of Liberal Arts) The Clar Structure of Fullerenes June 12, 2013 14 / 25

Improper Face 3-coloring Lemma The expansion E(C, A) is face 3-colorable. All faces of E(C, A) corresponding to faces in C and expanded edges of A will be in one color class. Furthermore, Γ has an associated improper face 3-coloring for which the only improperly colored faces are those that share edges of A. If the set C is in the blue color class, all Clar chains in Γ are yellow-red or red-yellow chains. d i d i ' d i f i-1 f i f i-1 ' t i ' f i ' t i+1 ' f i-1 f i g i g i ' g i Liz Hartung (Massachusetts College of Liberal Arts) The Clar Structure of Fullerenes June 12, 2013 15 / 25

Coxeter Coordinates To describe open Clar chains, we will use Coxeter coordinates. The distance between a pair of faces is given by their Coxeter coordinates. A shortest dual path between faces can be drawn as two line segments with a 120 left turn between them, or as one line segment. The Coxeter coordinates are given as (m, n), or just (m) in the case with no turn. Liz Hartung (Massachusetts College of Liberal Arts) The Clar Structure of Fullerenes June 12, 2013 16 / 25

Structure of non-interfering chains A straight chain segment is a chain of edges in A and hexagons f i such that the edges a i and a i+1 exit from opposite vertices of f i for each i. A straight chain segment with k edges in A connects a pair of faces with Coxeter coordinates (k, k). Every Clar chain in a fullerene Γ is a sequence of straight chain segments with only sharp turns. Liz Hartung (Massachusetts College of Liberal Arts) The Clar Structure of Fullerenes June 12, 2013 17 / 25

Non-interfering Chains We say that chains are non-interfering if they are separate from other pentagons over the fullerene. P 1 P 1 P 2 P 2 We can only connect two pentagons with a non-interfering Clar chain if they are in the same color class, or equivalently, if the Coxeter coordinates between them are congruent mod 3. Liz Hartung (Massachusetts College of Liberal Arts) The Clar Structure of Fullerenes June 12, 2013 18 / 25

Structure of non-interfering chains If two pentagons can be joined by a non-interfering Clar chain, then any shortest chain between them is composed of alternating right and left turns. Liz Hartung (Massachusetts College of Liberal Arts) The Clar Structure of Fullerenes June 12, 2013 19 / 25

Suppose that the segment between pentagons p 1 and p 2 has Coxeter coordinates (m, n) where m n and m n mod 3. The contribution to A depends on the position of the faces of C around the segment. P 1 P 1 P 2 P 2 If the set of yellow faces contains the set C, this chain will be of Type 1. A chain of Type 1 contributes m edges to A. Liz Hartung (Massachusetts College of Liberal Arts) The Clar Structure of Fullerenes June 12, 2013 20 / 25

Suppose that the segment between pentagons p 1 and p 2 has Coxeter coordinates (m, n) where m n and m n mod 3. P 1 P 1 P 2 P 2 This chain will be of Type 2 if the red faces contain the set C. A chain of Type 2 contributes m + n edges to A. Liz Hartung (Massachusetts College of Liberal Arts) The Clar Structure of Fullerenes June 12, 2013 21 / 25

Suppose that the segment between pentagons p 1 and p 2 has Coxeter coordinates (m, n) where m n and m n mod 3. P 1 P 1 P 2 P 2 If the blue faces in the figure contain the set C, the chain will be of Type 3. A chain of Type 3 contributes 3m + 2 edges to A. Liz Hartung (Massachusetts College of Liberal Arts) The Clar Structure of Fullerenes June 12, 2013 22 / 25

Closed Clar chains Because Clar chains do not cross one another, closed Clar chains must contain an even number of pentagons. A Clar structure that attains the Clar number won t have closed Clar chains that do not contain pentagons (we want to minimize A). Liz Hartung (Massachusetts College of Liberal Arts) The Clar Structure of Fullerenes June 12, 2013 23 / 25

Widely Separated Let Γ be a fullerene in which there is a non-interfering pairing of pentagons, and the Coxeter coordinates of the segments between each of the pairs are congruent mod 3. For each of the three possible choices for C over Γ, total the number of edges in A resulting for the six chains. Let M be the minimum of these three sums. We define this pairing of pentagons to be widely separated over the fullerene if for any two pentagons that are not paired together, one of the Coxeter coordinates of the segments joining them is at least M 2 2. Theorem Let Γ be a fullerene over which the pairs of pentagons are widely separated. Let M be the minimum sum of the edges of A over the three possible choices for the face set C. Then V 6 M 3 is the Clar number for Γ. Liz Hartung (Massachusetts College of Liberal Arts) The Clar Structure of Fullerenes June 12, 2013 24 / 25

Applications of Widely Separated Chains We can now use this method to compute the Clar number for classes of fullerenes in which the pairs of pentagons are widely separated. Graver s A Catalog of All Fullerenes with Ten or More Symmetries" provides examples for which we can compute the Clar number. It has generally been assumed that a set of independent benzene faces that attains the Clar number for a fullerene will be contained in the set of benzene faces that gives the Fries number. We construct an infinite family of fullerenes that serves as a counterexample. Liz Hartung (Massachusetts College of Liberal Arts) The Clar Structure of Fullerenes June 12, 2013 25 / 25