Some remarks on a paper of Chetwynd and Hilton on critical star multigraphs David Cariolaro Institute of Mathematics Academia Sinica Nankang, Taipei 11529 Taiwan E-mail: cariolaro@math.sinica.edu.tw June 30, 2008 Abstract In [A.G. Chetwynd and A.J.W. Hilton, Critical star multigraphs, Graphs and Combinatorics 2 (1986), 209-221] Chetwynd and Hilton started the investigations of the edge-chromatic properties of a particular class of multigraphs, which they called star multigraphs. A star multigraph is a multigraph such that there exists a vertex v ∗ that is incident with each multiple edge. Star multigraphs turn out to be useful tools in the study of the chromatic index of simple graphs. The main goal of this paper is to provide shorter and simpler proofs of all the main theorems contained in the above mentioned paper. We shall also give an extension of one of the Chetwynd-Hilton results to list-colouring and prove a general result of independent interest. Keywords: star multigraph, chromatic index, edge-colouring, fans MSC 2000: 05C15 1 Introduction All graphs considered in this paper are loopless, undirected and finite, but may contain multiple edges. The term “multigraph” will be used here as a synonym of “graph”. Let G be a graph. The vertex set and edge set of G are denoted by V (G) and E(G), respectively. The degree of a vertex v in G, denoted by dG (v), is the number of edges incident with v in G. ∆(G) denotes the maximum degree of G. If u, v are vertices of G, we denote by uv the set of edges joining u and v. The cardinality of uv is denoted by µ(uv) and called the multiplicity of uv. If µ(uv) = 1, we say that uv is a simple edge and, if µ(uv) > 1, we say that uv is a 1 multiple edge. A graph is simple if all its edges are simple. If e is an edge joining the vertices u and v, we denote this by e ∈ uv, or e = uv when uv is a simple edge. Two edges are adjacent if they are distinct and have at least one common endpoint, and parallel if they are distinct and have two common endpoints. If S is a set of vertices or edges of G, we denote by G − S the graph obtained from G by deleting all the elements of S, together with the edges incident to any vertex in S. An edge-colouring of G is a map ϕ : E(G) → C, where C is a set, called the colour-set, whose elements are called colours, and ϕ has the property that adjacent edges are assigned distinct colours. If C is chosen so that |C| is minimum, then ϕ is called an optimal colouring and the integer |C| is called the chromatic index of G, denoted by χ′ (G). G is said to be k-edge-colourable if k ≥ χ′ (G). Clearly χ′ (G) ≥ ∆(G), since all the edges incident with a vertex of maximum degree must receive a distinct colour. If χ′ (G) = ∆(G), we say that G is Class 1 and, otherwise, we say that G is Class 2. An edge e of G is called critical if χ′ (G − e) < χ′ (G). G itself is called critical if it is Class 2, has no isolated vertices, and all its edges are critical. It is well known and easy to see that every Class 2 graph G contains a critical subgraph with the same chromatic index. For an introduction to edge-colouring, and for graph-theoretic notation and terminology, not explicitly introduced here, we refer the reader to Fiorini and Wilson [7]. A celebrated theorem of Vizing [11] is equivalent to the statement that every Class 2 simple graph G satisfies χ′ (G) = ∆(G)+1. An extension of this theorem (Theorem 2 below) was first proved by Chetwynd and Hilton in [5], where the study of a new class of graphs, known as star multigraphs, was begun. A star multigraph is a graph G such that there exists a vertex v ∗ (called a star centre) to which all multiple edges of G are incident. Equivalently, G is a star multigraph if there exists a vertex v ∗ such that G − v ∗ is a simple graph. Thus a star centre may not be unique, but is unique unless G is a simple graph or there exists only one multiple edge in G. Star multigraphs, as stated in [5], are “vital tools in the investigations of the chromatic index of certain kinds of simple graphs”. The underlying philosophy is that, if the chromatic index of a certain simple graph H is unknown, it may be helpful to embed H into a star multigraph G (typically by the addition of a vertex v ∗ and suitably chosen multiple edges joining v ∗ to the vertices of H) in the attempt to draw information about H from the knowledge of the edge-colouring properties of G. It was this line of investigation that led Chetwynd and Hilton in [6] to formulate the Overfull Conjecture, which is now considered one of the most interesting and difficult conjectures in edge-colouring. Thus, it appears that star multigraphs offer a very fruitful line of investigation. In [5] Chetwynd and Hilton completely classified star multigraphs with at most two vertices of maximum degree. In [6] they classified certain star multigraphs with three vertices of maximum degree. (To the best of our knowledge the problem of the classification of all the star multigraphs with three vertices of maximum degree remains open to this date.) The proofs of the results contained in [6] are considerably more involved than those in [5], and we shall not 2 be concerned with them in this paper. Instead, we shall provide simplifications to the proofs of all the main results of [5], and sometimes our simplifications will be considerable. We remark that all the proofs in [5] and [6] are (in the style of the authors) very precise and, at times, ingenious. However the reading of [5] and [6] can prove to be difficult for those readers not accustomed with edge colouring, in particular for those parts of the proofs where fans or the so-called fan argument is used. One of our goals will be to provide some conceptual simplifications to the Chetwynd-Hilton proofs. Indeed, having put on a sound and formal basis the theory of fans in [2], we shall use some of our results to express, very succinctly, facts concerning fans in a rigorous way, in the hope to give to the reader a better understanding of the material presented. In particular, we shall frequently use an expression for the chromatic index of a Class 2 multigraph which we recently obtained [3, 4] (Lemma 1 below). We call this expression the Fan Formula. Quite unexpectedly, the Fan Formula gives an exact expression for the chromatic index of a Class 2 graph under very general conditions. Thus the paper may also be viewed as an attempt to show the power and wide applicability of this formula. Before we state it, however, we need to expose the reader to a certain amount of jargon. This will be compensated later by considerably shortening the proofs of the theorems. An e-tense colouring φ of a graph G is a partial edge-colouring of G which assigns no colour to e and whose restriction to E(G − e) is an optimal colouring of G − e. The colour set of φ is defined to be the colour set of its restriction to G − e. The edge e is called the uncoloured edge. Given an e-tense colouring φ of G with colour set C, and a vertex w ∈ V (G), we say that a colour α ∈ C is missing at w (or that w is missing the colour α) if there is no edge, having w as an endpoint, which is assigned the colour α by φ. Let e be an edge of G and let u be an endpoint of e. Let φ be an e-tense colouring of G. A fan at u with respect to φ is a sequence of edges of the form F = [e0 , e1 , e2 , . . . , ek−1 , ek ], where e0 = e, ei ∈ uvi , and where the vertex vi is missing the colour of the edge ei+1 , for every i = 0, 1, . . . , k − 1. The vertex u is called the pivot of the fan. The fan F is said to terminate at the edge ek . A fan is maximal if it cannot be extended to a larger fan. An edge f is called a fan edge at u if it appears in at least one fan at u. A vertex w is called a fan vertex at u if it is joined to u by at least one fan edge. The set of fan vertices1 is denoted by V (F). If w is a fan vertex at u, we denote by µ∗ (uw) the number of fan edges joining u and w, and call µ∗ (uw) the fan multiplicity of the edge uw. We are now ready to state the Fan Formula [3, 4]. Lemma 1 Let G be a Class 2 multigraph and let e ∈ uv be a critical edge. Let φ be a tense colouring with respect to the edge e, and let V (F) be the set 1 As the notation suggests, the set of fan vertices is the vertex set of a graph, which is called the Fan Digraph and was introduced and studied in [2]. However, this concept will not be necessary in the present context. We refer the reader to [2] for further details. 3 of fan vertices at u with respect to φ. Then χ′ (G) = 1 |V (F )| 1 = ⌈ |V (F )| · P · P w∈V (F ) (degG (w) w∈V (F ) (degG (w) + µ∗ (uw)) + |V (F )|−2 |V (F )| + µ∗ (uw))⌉ We shall say that the Fan Formula is written at u, to indicate that the pivot of the fans is the vertex u. Notice that |V (F)| ≥ 2 holds always under the hypotheses of Lemma 1, from which the second equality above follows easily. The Fan Formula may be deduced easily from the main theorem of [2], the Fan Theorem. We shall often use the following property, discovered independently by Andersen [1] and Goldberg [8, 9] and implicit in the work of Vizing (see [2, Lemma 2]). Lemma 2 Let G be a Class 2 graph, e a critical edge and φ an e-tense colouring of G. Let u be an endpoint of e. Let V (F) be the set of fan vertices at u. Then, for any two distinct vertices x, y ∈ V (F) ∪ {u}, and for any colour α missing at x and β missing at y, we have α 6= β. The exposition and the organization of the results of this paper does not follow the same order as in Chetwynd and Hilton [5]. Instead the paper is organized as follows. Section 2 is dedicated to star multigraphs with one vertex of maximum degree and to the proof that every Class 2 star multigraph G satisfies χ′ (G) = ∆(G) + 1. For a special class of star multigraphs we also determine the list-chromatic index. In Section 3 we prove two lemmas. The first of these is a general edgecolouring result which will be only of partial use here, but which may prove to be very helpful in the future. The second lemma specifies the edge-colouring properties of the complete graph with one edge replaced by r parallel edges, and will be very helpful in Section 4. Finally, in Section 4 we completely classify star multigraphs with two vertices of maximum degree. There are some additional results in [5] which we will not prove. Their proof is, in our view, sufficiently simple as it is in the original source, and most of them are corollaries of the results proved by us, or can be proved by an easy adaptation of the arguments adopted by us. 2 Star multigraphs with one vertex of maximum degree The first of our theorems will be stated in terms of list-colouring. We first give some relevant definitions. If G is a graph, an edge-list-assignment L is a function which assigns to each edge e of G a list L(e) of colours (i.e. a set). We say that G is L-choosable if it is possible to select a colour from each list and 4 assign it to the corresponding edge in such a way that the resulting colouring is a proper edge-colouring. If all the lists have cardinality k, we call L a klist-assignment. G is said to be k-edge-choosable if, for every k-list-assignment L, G is L-choosable. The minimum k for which G is k-edge-choosable is called list-chromatic index of G and denoted by χ′l (G). Clearly χ′l (G) ≥ χ′ (G), as may be seen by taking all the lists to be coincident with a fixed colour-set C, and the well known List-Colouring Conjecture asserts that χ′l (G) = χ′ (G) for any graph G. We are ready to prove our first theorem. Hilton and Chetwynd proved the edge-colouring part of the same theorem as part of their proof of [5, Theorem 1]. We shall generalize this result to its list-colouring analogue. In doing so we shall use the easy fact that the List-Colouring Conjecture holds for graphs G with ∆(G) ≤ 2 and a result of Harris [10] that says that, if a simple graph satisfies ∆(G) ≥ 3, then χ′l (G) ≤ 2∆(G) − 2. Theorem 1 Let G be a star multigraph such that every vertex is incident with a multiple edge. Then   ∆(G) + 1 if |V (G)| = 3, |E(G)| = ∆(G) + 1; χ′l (G) = χ′ (G) =  ∆(G) otherwise. Proof. If |V (G)| ≤ 3 the statement of the theorem is immediate, so we shall assume |V (G)| ≥ 4. Let v ∗ be the star centre (which is necessarily unique in this case) and let V (G) = {v ∗ , v1 , v2 , . . . , vs }, where s ≥ 3. Let µ(v ∗ vi ) = ki for all i = 1, 2, . . . , s. By assumption, ki ≥ 2, so v ∗ has degree ∆(G). Let L = {L(e) : e ∈ E(G)} be a ∆(G)-list-assignment. Choose a distinct colour from each of the lists assigned to the edges incident with v ∗ , thus obtaining a partial colouring ψ. We now aim to extend ψ to the edges of the simple graph H = G − v ∗ . Consider the list-assignment L1 = {L1 (e) : e ∈ E(H)}, where, if e = vi vj , L1 (e) is obtained from L(e) by suppressing the colours assigned by ψ to the edges of the form v ∗ vi and v ∗ vj . Clearly G is L-choosable if H is L1 -choosable. By construction, X |L1 (e)| ≥ |L(e)|−ki −kj = ∆(G)−ki −kj = kℓ ≥ 2(s−2) ≥ 2∆(H)−2, (1) ℓ6=i,j Ps where we have used the fact that dG (v ∗ ) = ℓ=1 kℓ = ∆(G) and s = |V (H)| ≥ ∆(H) + 1. By Harris’ theorem and (1), H is L1 -choosable as long as ∆(H) ≥ 3. We may then assume ∆(H) ≤ 2. If ∆(H) = 0, there is clearly nothing to prove. If ∆(H) = 1, then H is L1 -choosable unless there is an edge e ∈ E(H) such that L1 (e) = ∅. But, in view of (1), this may occur only if s = 2, which is contrary to assumption. If ∆(H) = 2 and H contains no odd cycle, then χ′l (H) = 2, and hence, from (1), H is L1 -choosable. If ∆(H) = 2 and H contains an odd cycle, then χ′l (H) = 3; in this case it suffices to show that at least one of the inequalities in (1) is strict. If s ≥ 4, the last inequality is strict. Hence we may 5 assume that s = 3. It is easy to see that, by a different initial choice for the partial list-colouring ψ, the first inequality in (1) may be assumed to be strict for all the edges of H. Hence, in any case, H may be assumed to be L1 -choosable and thus G is L-choosable, concluding the proof. ✷ Using Theorem 1, we can now prove that Vizing’s theorem for simple graphs extends to star multigraphs. Our proof is very short. Theorem 2 Let G be a star multigraph. Then χ′ (G) ≤ ∆(G) + 1. Proof. Without loss of generality, we may assume that G is Class 2. Furthermore, replacing G with a critical subgraph with the same chromatic index, we may assume that G is critical. If every vertex is incident with a multiple edge, then, by Theorem 1, G is (∆(G) + 1)-edge-choosable, and hence (∆(G) + 1)edge-colourable. Therefore we may assume the existence of a vertex u incident only with simple edges. Choosing an arbitrary edge e incident with u and any e-tense colouring φ, and writing the Fan Formula at u, we have P 1 ∗ χ′ (G) = ⌈ |V (F w∈V (F ) (degG (w) + µ (uw))⌉ )| · 1 ≤ ⌈ |V (F )| · P w∈V (F ) (degG (w) + 1)⌉ ≤ ∆(G) + 1, where we have used the fact that µ∗ (uw) = µ(uw) = 1 for any w ∈ V (F). This proves the theorem. ✷ By Theorem 2, any Class 2 star multigraph with maximum degree ∆ has necessarily chromatic index ∆ + 1. We proceed by considering star multigraphs with only one vertex of maximum degree. Theorem 3 Let G be a star multigraph with only one vertex of maximum degree. Then G is Class 2 if and only if G contains a subgraph on 3 vertices with ∆(G) + 1 ≥ 3 edges. Proof. If G contains a subgraph on 3 vertices as stated by the theorem then G is clearly Class 2. Assume now that G is Class 2. Replacing G with any of its critical subgraphs with the same chromatic index, we may assume that G is critical. If every vertex of G is incident with a multiple edge, then G, in view of Theorem 1, has necessarily the form prescribed by the theorem. Hence we may assume that G has a vertex u incident with simple edges only. Writing the Fan Formula at u (with respect to any edge e incident with u and any e-tense colouring φ), we have χ′ (G) = = 1 |V (F )| 1 |V (F )| · P ≤ ∆(G) + · P w∈V (F ) (degG (w) w∈V (F ) (degG (w) 1 |V (F )| + |V (F )|−2 |V (F )| + µ∗ (uw)) + + 1) + |V (F )|−2 |V (F )| = ∆(G) + 6 |V (F )|−2 |V (F )| |V (F )|−1 |V (F )| < ∆(G) + 1, where we have used the fact that at most one of the neighbours of u has degree ∆(G). Hence G is Class 1, contradicting the assumption. This contradiction proves the theorem. ✷ 3 Some Lemmas Before we continue our analysis of star multigraphs, we prove two lemmas in this section. The first lemma expresses an edge-colouring property of arbitrary critical multigraphs. A small application of this lemma will be given in the proof of Lemma 2. The result itself, however, is of independent interest. It may prove to be useful in the study of such conjectures as the Overfull Conjecture or the Goldberg-Seymour Conjecture, as well as in the further study of star multigraphs. We shall use the following notation: if G is a graph and x, y are vertices of G, we denote by G + xy the graph obtained from G by adding an edge joining x to y. Lemma 3 Let G be a critical graph with more than three vertices and let x, y be adjacent vertices. Then χ′ (G + xy) = χ′ (G). Proof. Assume, on the contrary, that χ′ (G + xy) > χ′ (G). First observe that every edge in G is incident with either x or y (or both). For, assume there was an edge h neither incident with x nor with y. Since G is critical, χ′ (G−h) < χ′ (G), so there is an optimal colouring ϕ of G and a colour α such that ϕ−1 (α) = {h}, i.e. h is the only edge receiving colour α. Such colouring is easily extendable to a colouring of G + xy by colouring the extra edge joining x, y with colour α, thus contradicting the assumption that χ′ (G + xy) > χ′ (G). Hence every edge is incident with either x or y, i.e. the graph G − xy is bipartite with bipartition ({x, y}, V (G) \ {x, y}). Let φ be any optimal colouring of G. We may think of φ as a tense colouring of G + xy, where the uncoloured edge is the extra edge joining x and y. Let e be any edge joining x and y in G. Let ǫ = φ(e). Since every edge of G is adjacent or coincident with e, there is no other edge in G coloured ǫ, and hence the colour ǫ is missing at every vertex other than x and y. Let α be a colour missing at x under the colouring φ and let β be a colour missing at y. If α = β, then we can colour the uncoloured edge with colour α, thus contradicting the assumption that χ′ (G + xy) > χ′ (G). It follows that α 6= β. There must be a bicoloured α-β path joining x and y, otherwise a colour exchange along a bicoloured α-β path starting at x would result in a colouring φ′ such that x and y are missing the same colour under φ′ , and this would contradict what was proved above. The bicoloured α-β path joining x and y, in the present case, has necessarily length 2, and hence is of the form xzy. If all the χ′ (G) colours used by φ appeared on one of the edges xy, yz, xz, then G would have a subgraph on 3 vertices with the same chromatic index as G, contradicting the fact that G is critical and |V (G)| ≥ 4. Hence there exists a colour λ which is not present on the edges of the multitriangle xyz. By symmetry, we may assume that λ is present at the vertex x, say on the edge 7 xw, where w 6= y, z. The vertex z is a fan vertex at x with respect to φ, and is missing colour ǫ. Since w is also missing the colour ǫ under the colouring φ, it cannot be a fan vertex at x, and hence the colour λ must be present at y. It follows the existence of an edge of the form yt, where t 6= x, w, z, which is coloured λ. Now, the colour β is present at at most one of w, t (since it is present at z and there may be at most two edges coloured β). Suppose β is missing at w. Then, by interchanging the colours of the bicoloured λ-β path wxz, we may guarantee that β is missing at z. But now both z and w become fan vertices at x under the current colouring φ′ , because [e0 , e1 , e2 ] is a fan at x, where e0 is the uncoloured edge, e1 is the λ-edge joining x to z, and e2 is the β-edge joining x to w. But z and w are both missing the same colour ǫ under φ′ , which contradicts the fact that they are both fan vertices at x. Therefore we may assume that β is missing at the vertex t under the colouring φ. However, interchanging the colours of the edges of the α-β path joining x and y yields a colouring φ′′ such that β is missing at x and α is missing at y, and hence creates a situation symmetrical to the one of the other case, which also results in a contradiction. This contradiction proves the lemma. ✷ We now prove a simple lemma which will be helpful in the sequel. For positive integers n, r, we2 denote by Kn+r the complete graph Kn with one edge replaced by an edge of multiplicity r. It is obvious that K3+r is Class 2 for every +r r, and easy to see that K2n is Class 1 for every r and n. Indeed +r +r χ′ (K2n ) ≤ χ′ (K2n ) + r − 1 = 2n + r − 2 = ∆(K2n ). The following lemma deals with the remaining cases. We shall use the fact that K2n+1 is Class 2 for every n and, in each optimal colouring of K2n+1 , every vertex is missing a distinct colour. Lemma 4 Let r, n be positive integers, n ≥ 2, r ≥ 2. Then the following are equivalent properties: (a) r = 2; +r (b) K2n+1 is Class 2; +r (c) K2n+1 is critical. +2 Proof. Assume r = 2. We prove that K2n+1 is Class 2. This is immediate +2 since, if K2n+1 was Class 1, it would have a (2n + 1) edge-colouring; but then we would have a (2n + 1)-edge-colouring of K2n+1 with two vertices missing +2 the same colour, which is impossible. Hence K2n+1 is Class 2. We now prove +2 that it is critical. Let xy be the multiple edge of K2n+1 . Removing one of the +2 two parallel edges yields the graph K2n+1 and, since K2n+1 is Class 2, we have +2 +2 χ′ (K2n+1 ) > 2n + 1 = χ′ (K2n+1 ). Remove now any other edge of K2n+1 , say 2 The +2 + corresponding notation for the graph Kn in the Chetwynd-Hilton paper is Kn . 8 e, and consider the graph K2n+1 − e. It is easy to construct a (2n + 1)-edgecolouring of K2n+1 − e with the two vertices x, y missing the same colour α, and hence it is possible to colour with colour α an additional edge joining x and y, +2 +2 +2 proving the inequality χ′ (K2n+1 − e) = 2n + 1 < χ′ (K2n+1 ). Hence K2n+1 is critical. Assume now r = 3. By Lemma 3 (using the assumption n ≥ 2) we have +3 +2 +3 χ′ (K2n+1 ) = χ′ (K2n+1 ) = 2n + 2 = ∆(K2n+1 ), +3 and hence K2n+1 is Class 1. It follows by an easy induction argument that, for +r every r ≥ 4, K2n+1 is Class 1, since +(r−1) +r +r χ′ (K2n+1 ) ≤ χ′ (K2n+1 ) + 1 = 2n + r = ∆(K2n+1 ). Thus the proof is completed. 4 ✷ Star multigraphs with two vertices of maximum degree We are now ready to prove the following theorem. Theorem 4 Let G be a star multigraph with two vertices of maximum degree, one of which a star centre. Then G is Class 2 if and only if G contains a subgraph +2 on 3 vertices and ∆(G) + 1 edges or G = K2n+1 , for some n ≥ 2. Proof. If G has the form prescribed by the theorem then clearly G is Class 2. Assume now that G is Class 2. Let H be a critical subgraph of G with the same chromatic index (and hence with the same maximum degree). Arguing by contradiction, we assume that H has more than 3 vertices. By Theorem 3 and the assumption of criticality, H must have two vertices of maximum degree, and hence H and G have the same vertices of maximum degree. Let v ∗ be a star centre. If there is a vertex u of H not adjacent to v ∗ , then, using the Fan Formula at u and the fact that u is adjacent to at most one vertex of maximum degree of H, we have P |V (F )|−2 1 ∗ χ′ (G) = |V (F w∈V (F ) (degG (w) + µ (uw)) + |V (F )| )| · = 1 |V (F )| · P ≤ ∆(G) + w∈V (F ) (degG (w) 1 |V (F )| + |V (F )|−2 |V (F )| + 1) + |V (F )|−2 |V (F )| = ∆(G) + |V (F )|−1 |V (F )| < ∆(G) + 1, whence a contradiction with the fact that H is Class 2. It follows that every vertex is either adjacent or coincident with v ∗ . Let V (H) = {v ∗ , v1 , v2 , . . . , vs }, where dH (v ∗ ) = dH (u) = ∆(H), and let ki = µ(v ∗ vi ) for every i = 1, 2, . . . , s. Then s s X X ki (2) ki = k1 + ∆(H) = i=2 i=1 9 and dH (v1 ) = k1 + t = ∆(H), (3) ∗ where t is the number of neighbours of v1 in J = H − v . By comparison of (2) and (3), we see that t=s−1 (4) and ki = 1 for every i = 2, 3, . . . , s. (5) H ⊂ Kn+r (6) ∆(H) = ∆(Kn+r ) (7) It follows that and for some n ≥ 4 and r ≥ 2. Since H is Class 2, it follows from (6) and (7) that Kn+r is Class 2 and, by Lemma 4, that n is odd, n ≥ 5 and r = 2. That is to +2 +2 say, we have H ⊂ K2n+1 for some n ≥ 2 and ∆(H) = ∆(K2n+1 ) = 2n + 1. +2 +2 Since K2n+1 is critical and H is Class 2 and ∆(H) = ∆(K2n+1 ), we conclude +2 that H = K2n+1 . Hence every vertex of H has degree 2n, except v ∗ and v1 , which have degree 2n + 1. The addition of any edge incident with any vertex of H would therefore either increase the maximum degree of H or increase the number of vertices of maximum degree in H. Since H has the same maximum degree and the same number of vertices of maximum degree as G, it follows that G = H, and the proof is completed. ✷ To complete the classification of star multigraphs with two vertices of maximum degree we now prove the following. Theorem 5 Let G be a star multigraph with two vertices of maximum degree, neither of which a star centre. Then G is Class 1. Proof. We argue by contradiction, so let us assume that G is Class 2. Replacing G by a critical subgraph with the same maximum degree (which must necessarily satisfy the same conditions) we may assume that G is critical. Let v ∗ be a star centre and let v1 , v2 be the vertices of maximum degree. By assumption, v ∗ 6= v1 , v2 . If every vertex of G was adjacent or coincident to v ∗ , then v ∗ would have maximum degree in G, contrary to assumption. Hence there exists a vertex u which is not adjacent to v ∗ . Writing the Fan Formula at u with respect to any edge e incident with u and any e-tense colouring, we see that P |V (F )|−2 1 ∗ χ′ (G) = |V (F w∈V (F ) (degG (w) + µ (uw)) + |V (F )| )| · = 1 |V (F )| · P ≤ ∆(G) + w∈V (F ) (degG (w) 2 |V (F )| + |V (F )|−2 |V (F )| + 1) + |V (F )|−2 |V (F )| = ∆(G) + 1, 10 where we have used the fact that there are at most two neighbours of u of maximum degree. Since the sign of equality must hold in the last inequality above, both v1 and v2 are fan vertices at u, and in particular u is adjacent to both v1 and v2 and hence is distinct from v1 and v2 (which implies that both v1 and v2 are adjacent to v ∗ , otherwise, replacing u with one of them and repeating the argument, we reach a contradiction). Now3 , deleting the edge uv2 and adding an edge joining u to v ∗ , we obtain a star multigraph G′ with star centre v ∗ and with ∆(G) = ∆(G′ ). Let ∆ = ∆(G) = ∆(G′ ). Notice that G′ has at most two vertices of degree ∆, namely v1 and (possibly) v ∗ . Suppose G′ was Class 2. Then, by Theorem 3 and Theorem 4, either G′ contains a subgraph on 3 vertices and ∆ + 1 edges (which is impossible, since this would force v ∗ to have only two neighbours in G′ , whereas has the three neighbours u, v1 , v2 ), +2 or G′ = K2n+1 for some n ≥ 2 (which is also impossible, since G′ contains two non-adjacent vertices, namely u and v2 ). We conclude that G is Class 1. Thus there exists a ∆-edge-colouring of G′ , and hence there exists a ∆-edge-colouring φ of G′ − uv ∗ = G − uv2 such that u and v ∗ are missing the same colour α. We can view φ as an e0 tense colouring of G, where e0 = uv2 is the uncoloured edge. Then u is a fan vertex at v2 missing colour α and, consequently, v ∗ cannot be a fan vertex at v2 , since it is also missing colour α. Since dG (u) < ∆, there are at least two missing colours at u, say α and β, and two maximal fans F1 , F2 , one of which of the form F1 = [e0 , e1 , . . .], where e1 is coloured α, and the other of the form [e0 , f1 , . . .], where f1 is coloured β. Since no fan vertex at v2 is joined to v2 by a multiple edge, it follows that all the edges of F1 (and, similarly, F2 ) are incident with distinct fan vertices, and the fan vertices of F1 are distinct from those of F2 , except for u, which is an endpoint of e0 . Moreover both F1 and F2 must terminate with a fan edge incident with a vertex of maximum degree, which is necessarily v1 . Hence F1 and F2 both contain v1 as a fan vertex, contradicting the fact that the only common fan vertex is u, and this contradiction establishes the theorem. ✷ Final Remarks: In this paper we have obtained simplifications of the proofs of the main results of [5] by means of some tools related to the study of fans, in particular the Fan Formula. Such tools may be applied to classes of graphs other than star multigraphs. Lemma 3 may have applications which go beyond the scope of this paper. In particular it may be useful in order to settle the classification problem for star multigraphs with three vertices of maximum degree. References [1] L.D. Andersen, On edge-colourings of graphs, Math. Scand., 40 (1977), 161-175. 3 The following ingenious argument is the same used by Chetwynd and Hilton in [5, Lemma 15]. However the present proof is considerably shorter because we do not prove, as a preliminary step, that ∆(G) ≤ |V (G)| − 1, which is unnecessary in our argument. 11 [2] D. Cariolaro, On fans in multigraphs, J. Graph Theory, 51 (2006), 301-318. [3] D. Cariolaro, Some consequences of a theorem on fans, to appear in Taiwanese J. Mathematics. [4] D. Cariolaro, The fire index, Graph Theory Notes of New York, LIV:1 (2008), 9-14. [5] A.G. Chetwynd and A.J.W. Hilton, Critical star multigraphs, Graphs and Combin., 2 (1986), 209-221. [6] A.G. Chetwynd and A.J.W. Hilton, Star multigraphs with three vertices of maximum degree, Math. Proc. Cambr. Phil. Soc., 100 (1986), 303-317. [7] S. Fiorini and R.J. Wilson, Edge-Colourings of Graphs, Research notes in Mathematics, Pitman, 1977. [8] M.K. Goldberg, Remark on the chromatic class of a multigraph, Vycisl. Mat. i Vycisl. Tech. (Kharkov), 5 (1975), 128-130 (in Russian). [9] M.K. Goldberg, Edge-coloring of multigraphs: recoloring technique, J. Graph Theory, 8 (1984), 123-137. [10] A.J. Harris, Problems and conjectures in extremal graph theory, Ph.D. thesis, Cambridge University, 1984. [11] V.G. Vizing, On an estimate of the chromatic class of a p-graph, Diskret. Analiz., 3 (1964), 25-30 (in Russian). 12