Let v be a vertex primed by v i (s). Then the number f(v) of neighbours of v which have

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1 Let v be a vertex primed by v i (s). Then the number f(v) of neighbours of v which have been red in the sequence up to and including v i (s) is deg(v)? s(v), and by the induction hypothesis this sequence is an initial segment in < T. By the denition of s, f(v) is equal to jn T (v)j, where N T (v) contains the neighbours x of v which satisfy x < T v 0 or x = v 0, that is, another initial segment in < T. Hence the f(v) vertices which have been red are precisely those in N T (v). In particular, the last of these f(v) vertices in the order < T is v i (s), which must therefore be v 0. This proves that v is adjacent to v i (s) in T, so that S i+1 is a subtree of T. To complete the induction step, it remains to show that if u is the vertex which is next after v i (s) in < T, then it is also the next vertex in < s ; that is, u = v i+1 (s), the next vertex red. All vertices which precede u in < T have been red, and these include all vertices in N T (u). Thus, by the denition of s = s T, u is ready. We have already shown that S i+1 is a subtree of T, so u must be next after v i (s) in the induced order on S i+1, and by the rules for FC, u = v i+1 (s). The combined eect of the foregoing Theorems is to show that the mappings s 7! T s and T 7! s T are inverse bijections. Furthermore, if s and T correspond under these mappings, then the associated orderings < s and < T are the same. References 1 Biggs, N.L. Algebraic Graph Theory, Second Edition. Cambridge University Press, Biggs, N.L. Algebraic Potential Theory on Graphs. Bulletin of the London Mathematical Society, to appear. 3 Biggs, N.L. Chip-ring and the critical group of a graph. CDAM Research Report Series, LSE-CDAM-96-03, January (Submitted for publication.) 4 Bjorner, A., Lovasz, L., and Shor, P. Chip-ring games on graphs. Europ. J. Combinatorics 12 (1991) Lovasz, L. and Winkler, P. Mixing of random walks and other diusions on a graph. In: Surveys in Combinatorics 1995, ed. P. Rowlinson. Cambridge University Press, 1995, Tutte, W.T. A contribution to the theory of chromatic polynomials. Canad. Math. J. 6 (1954)

2 Theorem 1 Given a critical conguration s, the sequence v 0 (s); v 1 (s); : : : ; v n?1 (s) de- nes a total ordering < s on V, and S n is a spanning tree T s of G. We have to show that, for 1 i < n, S i contains a vertex which is ready. It is Proof enough to show that some vertex v 6= q is ready in the conguration s i, because all such vertices have been included in some S j, j i, which is a subtree of S i. We are given that s is recurrent, so there is a q-legal sequence of rings starting from s which produces s again; also there is a q-legal sequence starting from s which produces s i. Hence, by the conuence property [3], there is a q-legal sequence which leads from s i to s, which means that some vertex must be ready in s i. The second construction Given a spanning tree T of G, the second construction produces a total ordering < T inverse to FC. T SC 7?! (< T ; s T ) of V and a critical conguration s T. Furthermore, SC is The rst part of SC depends on the simple observation that given T, and any vertex v 2 V, there is a unique path in T from q to v. We can therefore dene an order < T on the vertices of G by using the order on the paths. Theorem 2 Suppose that a conguration s is given and T = T s is the spanning tree obtained by FC. Then the ordering < T dened by SC is the same as the ordering < s dened by FC. Proof By denition, q comes rst in both orderings. Consider the relationship between v i?1 (s) and v i (s) in FC. There are two possibilities: either v i (s) is primed by v i?1 (s), or it is not. In the rst case the edge joining the two vertices is in S i, and since S i is a subtree of T s = T, it is in T. It follows that v i?1 (s) < T v i (s). In the second case, v i (s) is ready before v i?1 (s) is red, and so both vertices are ready in s i?1. Both vertices are in S i?1, and since v i?1 (s) was red before v i (s), it must come before v i (s) in the induced order on the vertices of S i?1, and hence in the induced order on T. For each vertex v 6= q let v 0 be the unique vertex adjacent to v on the path from q to v in T. Dene N T (v) = fv 0 g [ fx 2 V j x < T v 0 and xv 2 Eg: The conguration s T is dened by s T (v) = deg(v)? jn T (v)j. Theorem 3 Suppose that a spanning tree T is given, and s = s T is as dened above. Then the spanning tree T s produced by FC is the same as T. Proof By FC, T s = S n. We make the induction hypothesis that S i is a subtree of T, and the sequence v 0 (s); v 1 (s); : : : ; v i (s) is an initial segment of the order < T. Clearly this holds when i = 0.

3 In summary: any set of b 2 rim edges determines a unique conguration with external activity 0 and internal activity b; when b = 1 any rim edge except e p+1 determines such an s. It follows that the number of congurations with internal activity b and external activity 0 is t b =? p b if b 2; p? 1 if b = 1. From this we can obtain the chromatic polynomial of W p, the wheel with n = p+1 vertices, using the general formula for P (G; ) given in Section 4: n?1 X i=1 (? 1)? p p p? 1 = (?1) n?1?i t i (? 1) i (? 1) p?1 + : : : (?1) p p 2 (? 1) 2 + (?1) p+1 (p? 1)(? 1) Comparing with the binomial expansion of ([? 1]? 1) p we obtain the result P (W p ; ) = ((? 2) p + (?1) p (? 2)) : : We assume the notation set up in Section 3. The rst construction The rst construction (FC) is an assignment APPENDIX s 7?! F C (< s ; T s ); which produces a total ordering < s of V (the standard ordering, in the terminology of Section 3) and a spanning tree T s of G. It involves a sequence v 0 (s); v 1 (s); : : : ; v n?1 (s); of vertices of G, together with a sequence s 0 ; s 1 ; : : : ; s n = s of congurations and a sequence S 0 ; S 1 ; : : : ; S n of subtrees of G. Set v 0 (s) = q, s 0 = s, and S 0 = fqg. For i = 1; 2; : : : ; n take s i to be the conguration obtained from s i?1 by ring v i?1 (s). Then take S i to be the union of S i?1 and the vertices which are primed by v i?1 (s), together with the edges joining these vertices to v i?1 (s). Since a vertex can be primed only once S i is a tree, and it contains q. Its vertices are therefore ordered by the order on the paths in S i joining them to q. Thus we can take v i (s) to be the rst (in this order) vertex of S i which is ready, provided that there is such a vertex.

4 (a)! > + 1 (b)! = + 1 Figure 2: typical congurations with external activity 0 The exception is that, provided that! > + 1, the vertex! is not primed by q or by! + 1, but by the third of its neighbours to be red,!? 1. Thus s(!) = 0 (Figure 2a). In the case! = + 1, we have already established that s( + 1) = 2, and the situation is as shown in Figure 2b. The internal activity of s is easily calculated by considering each type of internal edge. Clearly e 1 is active. Let e +1 be any other spoke which is internal. If e +1 is in (e ) where e is an external spoke, then Figure 2 shows that > + 1; if e +1 is in (e n+ ) where e n+ is an external rim edge, it is automatic that > +1. Hence e +1 is active. Let e n+ ( 6= n) be a rim edge which is internal. Then e +1 is external. Since e n+ 2 (e +1 ) and + 1 < n +, e n+ is not active. (If = n, e 1 is external, and so on.) It follows that the internal activity of any critical conguration s with ext(s) = 0 is just the number b of internal spokes, Conversely, suppose we are given any set of b rim edges. Then we can use the construction illustrated in Figure 2 to dene a conguration s for which these are precisely the external rim edges. We shall show that the external activity of s is zero except in one case. Suppose rst that b 2. If e is an external spoke, the path (e ) contains the preceding internal spoke, so no spoke is externally active. If e p+ is any external rim edge except the last, (e p+ ) contains e +1, so e p+ is not active. Finally, (e p+! ) contains e 1. Hence the external activity is zero whenever b 2. If b = 1, e p+! is the only external rim edge, and (e p+! ) consists of all the other rim edges, so e p+! is not externally active unless it is e p+1.

5 Figure 1: proof that e +1 is not external Suppose that the external rim edges are (in order) e p+ ; e p+ ; : : : ; e p+ ; e p+!. Let e p+ be any one of them, except the last ( 6=!). We shall show that the spoke e +1 is internal for s. Suppose, for a contradiction, that e +1 is external (Figure 1). Then the edge e p++1 must be internal, since it is the only remaining edge which can be used to prime the vertex + 1. Now the path (e +1 ) must consist of a sequence of consecutive rim edges starting with e p++1, followed by a spoke. It cannot contain the rim edge e p+!, since that is external, and so the spoke must be some e j with + 1 < j!. Thus (e +1 ) contains only edges e k with k > + 1, so e +1 is active, which is the required contradiction. The preceding observation enables us to characterise congurations s with ext(s) = 0. Each of the external rim edges e p+ except e p+! determines an internal spoke e +1, and (because there are b?1 of them) these together with e 1 must form the entire set of internal spokes. The internal spokes correspond to the vertices primed by q, so s(1) = s( + 1) = : : : = s( + 1) = 2: If = 1 then we already know that s() = 2. If > 1 there are internal rim edges e n+1 ; e n+2 ; : : : ; e n+?1, which means that 1 primes 2, 2 primes 3,..., and? 1 primes. In other words the vertices 2; 3; : : : ; are primed when the second of their neighbours is red, so s(2) = s(3) = : : : = s() = 1. For the same reason, all the remaining rim vertices x have s(x) = 1, except for one case.

6 5. The activity of critical congurations We could dene the internal and external activity of a critical conguration s by using the construction s 7! T s described in the Appendix, and applying the denitions given above to T s. However, the aim is to be more direct. Let s be a critical conguration, and < s the standard ordering dened in Section 3. Let us say that an edge xy is internal for s if either x primes y or y primes x in this process. If neither holds, we say xy is external for s. As we observed, given any vertex v there is a unique path (v) from q to v consisting entirely of internal edges. It follows that, given any two vertices v and w, there is a unique path (vw) containing those edges which are in exactly one of (v) and (w); if v and w are the ends of an external edge e, we call this path (e). The basic denitions can now be formulated as follows. An external edge e j is active if, for any edge e i such that e i 2 (e j ), it follows that i > j. An internal edge e i is active if for any edge e j such that e i 2 (e j ), it follows that i < j. We dene the internal (external) activity of s to be the number of internal (external) edges which are active. We denote these quantities by int(s) and ext(s) respectively. It is easy to check that, with these denitions, int(s) is the internal activity of the spanning tree T s, and ext(s) is the external activity of T s, where T s is the tree of internal edges for s as in the Appendix. But we can use the denitions without having to construct T s. It follows that we can take the number t i dened in Section 4 to be the number of critical congurations which have internal activity i and external activity Application to chromatic polynomials of wheels The results described above are independent of the vertex q and the ordering of the edge-set E; this means that we may choose them in any way appropriate to a specic case. For example, let W p be the wheel graph on p + 1 vertices (p 3). The vertices consist of p rim vertices which form a p-cycle, and one hub vertex which is joined to the rim vertices by edges which we call spokes. It seems appropriate to choose q to be the hub, and to order the edges so that the spokes are labelled e 1 ; e 2 ; : : : ; e p in cyclic order, and the edges joining the rim vertices are labelled e p+1 ; e p+2 ; : : : e 2p, also in cyclic order. We shall denote by i the rim vertex incident with the spoke e i (1 i p), and specify that the rim edges incident with 1 are e 2p and e p+1. As we shall see, these choices enable us to obtain a simple characterisation of the critical congurations which have external activity 0 and internal activity i, which leads to an explicit determination of the chromatic polynomial. Suppose that s is a critical conguration on (W p ; q), such that ext(s) = 0. Then e 1 cannot be external for s, because if it were it would be active. Thus e 1 is an internal spoke. Suppose there are b 1 internal spokes; then since p edges in all are internal for s, there must be p? b rim edges which are internal, and hence b rim edges which are external.

7 Let v 0 (s) = q and let s 0 be the given conguration s. For i = 1; 2; : : : ; n we take s i to be the conguration obtained from s i?1 by ring v i?1 (s). We shall say that a vertex x is primed by v i?1 (s) if x is ready in s i but not in s i?1. Any vertex y which is ready in s i has been primed at some stage, by the ring of a vertex which must itself have been primed, and so on. It follows that, for each such vertex y, there is a unique path (y) whose vertices q; y 1 ; y 2 ; : : : ; y k = y are such that q primes y 1 which primes y 2, and so on. We dene v i (s) to be the vertex y which is ready in s i and for which (y) is rst in the order on P induced by. In the Appendix it is proved that this procedure works; that is, the vertex v i (s) can be dened for i n? 1, and all these vertices are distinct. Thus we have a standard ordering of V, which we denote by < s. In addition, the set of edges such that one end primes the other form a spanning tree T s. 4. The chromatic polynomial and spanning trees Let z be a positive integer. The chromatic polynomial P (G; ) is a polynomial function which, when evaluated at = z, gives the number of ways of assigning one of z colours to each vertex of G, in such a way that adjacent vertices receive dierent colours. It is well-known [1, 6] that P (G; ) is a monic polynomial of degree n, and that P (G; 0) = 0. It follows that P (G; ) can be written in the form? t n?1 (? 1) n?1? t n?2 (? 1) n?2 + : : : + (?1) n t 1 (? 1) ; where t n?1 = 1. The signicance of this form of expansion is that the coecients t 1 ; t 2 ; : : : ; t n?1 can be interpreted as the numbers of spanning trees of certain kinds in G. Let T be a spanning tree. For each edge g 2 T there is a unique cut consisting of all the edges which have one in end in each of the components obtained by deleting g from T. We denote this by cut(t; g); it contains g itself and edges which are not in T. For each edge h which is not in T there is a unique cycle consisting of h and edges which are in T ; we denote this by cyc(t; h). Note that if k 2 T, l =2 T, then k is in cyc(t; l) if and only if l is in cut(t; k). In order to dene t i, we must assume (as in the construction of the standard ordering) that the edges of G are given a xed ordering e 1 ; e 2 ; : : : ; e m. Suppose e i 2 T. Then we say that e i is internally active if i is the least index of any edge in cut(t; e i ). Similarly, if e j =2 T, we say that e j is externally active if j is the least index of any edge in cyc(t; e j ). The internal (external) activity of T is dened to be the number of edges which are internally (externally) active. The fundamental result of Tutte [6] is: t i is the number of spanning trees which have internal activity i and external activity 0.

8 Let us say that vertex v 6= q is ready in a conguration s if s(v) deg(v), the degree of v. The government q is said to be ready just when no other vertex is ready. If v is ready in s, then it can be red, which results in a new conguration s 0 dened by 8< : s(x) + 1; if xv 2 E; s 0 (x) = s(x)? deg(v); if x = v; s(x); otherwise. If no vertex v 6= q is ready in s, then we say that s is stable. In a stable conguration q is the only vertex which can be red. Given any conguration s we say that a sequence of vertices v 1 ; v 2 ; : : : ; v k is q-legal for s if v 1 is ready in s, v 2 is ready in the conguration obtained from s by ring v 1, and so on; furthermore, we insist that q occurs if and only if the preceding conguration is stable. Since we are allowed to re q when there is no alternative, a q-legal sequence can continue indenitely. This means that there will be congurations s which are recurrent that is, there is a nite q-legal sequence of rings which starts and nishes with s. We say that a conguration is critical if it is both stable and recurrent. There is an interesting theory of critical congurations, based upon the `conuence' properties of q-legal sequences. For our present purposes the following result [3] is needed. Let s be any critical conguration on (G; q). Since s is recurrent there is a q-legal sequence of rings under which it recurs, and (crucially) there is a such a sequence in which each vertex occurs just once. Since s is stable, q must be the rst vertex in this sequence. The conditions stated do not determine a unique sequence but, as shown in the Appendix, if we are given some arbitrary (but xed) ordering of the edges of G, then there is a construction which produces a `standard' ring order under which s recurs. The details are outlined in the next section. 3. The standard ordering Let G = (V; E) be a simple connected graph, and set jv j = n, jej = m. We x once and for all a vertex q 2 V and an ordering e 1 ; e 2 ; : : : ; e m of E. Let P be the set of paths in G which begin at q. Since G is simple a path is specied by a sequence of edges, and there is an induced lexicographic ordering of P, dened as follows. If = ( 1 ; 2 ; : : : ; j ) and = ( 1 ; 2 ; : : : ; k ) are sequences of edges corresponding to paths in P and i is the least index for which i 6= i, then if and only if i comes before i in the order on E; if there is no such index then if and only if j < k. (For completeness, the empty set of edges is regarded as a path from q to q, and it comes rst in the order.) We can now specify the rule which, given any critical conguration s on (G; q) produces a standard sequence v 0 (s); v 1 (s); : : : ; v n?1 (s); containing all the vertices of G, and under which s recurs.

9 1. Introduction Chip ring and the chromatic polynomial Suppose that each vertex of a graph has a number of dollars, except for one vertex q, called \the government", which is in debt by the total amount of dollars held by the rest. A transaction (usually called ring a vertex v) consists of transferring one dollar from v to each of its neighbours. It is clear that a vertex v 6= q can only be red if it has at least as many dollars as it has neighbours, but this restriction does not apply to q, because ring q merely increases its debt. This is a variant of what is usually called a chip-ring game on the graph [4, 5]. It is known [2, 3] that the number of congurations of this game which are `critical', in a sense dened below, is equal to the number of spanning trees of the graph. The fact that the two sets have the same cardinality raises the question of an explicit bijection between them, but there are good reasons why there can be no `natural' bijection. For one thing, the set of critical congurations is an abelian group, and has a distinguished element, the identity in the group, whereas there is no correspondingly distinguished spanning tree. However, we have shown that, if the elements of the graph are ordered in some way, then there is a meaningful way of constructing a bijection. Two constructions, with proofs, are given in the Appendix to this paper. It is well-known that the chromatic polynomial of a graph can be expressed as a sum of contributions, each determined by a spanning tree of the graph. Thus our bijections enable us, in theory, to express the chromatic polynomial in terms of critical congurations: for each critical conguration we can construct the corresponding spanning tree, and calculate its contribution to the chromatic polynomial. In this paper we set out to establish a more direct link between critical congurations and the chromatic polynomial. We shall show that it is possible to dene the relevant quantities and carry out the calculations using the congurations themselves, rather than the associated spanning trees. The process will be illustrated by calculating the chromatic polynomial of a wheel. Of course, there are other ways of doing this particular calculation: our method may appear complicated, but that might be simply because it is unfamiliar. 2. Critical congurations The `dollar game' described informally in the Introduction can be dened as follows. Full details and proofs may be found in [3]. Let G = (V; E) be a graph which is simple (no loops or multiple edges), and let q 2 V be xed. A conguration on (G; q) is an integer-valued function s dened on V, such that s(v) 0 (v 6= q); s(q) =? X v6=q s(v):

10 January 1997 Chip-ring and the chromatic polynomial Norman Biggs London School of Economics Peter Winkler Bell Laboratories