EDGE-CHOOSABILITY OF CUBIC GRAPHS AND THE POLYNOMIAL METHOD

Transcription

1 EDGE-CHOOSABILITY OF CUBIC GRAPHS AND THE POLYNOMIAL METHOD by Andrea Marie Spencer B.MATH., University of Waterloo, 8 a Thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in the Department of Mathematics c Andrea Marie Spencer SIMON FRASER UNIVERSITY Spring All rights reserved. However, in accordance with the Copyright Act of Canada, this work may be reproduced, without authorization, under the conditions for Fair Dealing. Therefore, limited reproduction of this work for the purposes of private study, research, criticism, review and news reporting is likely to be in accordance with the law, particularly if cited appropriately.

2 APPROVAL Name: Degree: Title of Thesis: Andrea Marie Spencer Master of Science Edge-Choosability of Cubic Graphs and the Polynomial Method Examining Committee: Dr. John Stockie, Professor of Mathematics Chair Dr. Luis Goddyn, Senior Supervisor, Professor of Mathematics Dr. Ladislav Stacho, Supervisor, Professor of Mathematics Dr. Matthew DeVos, Internal/External Examiner, Professor of Mathematics Date Approved: April 7, ii

3 Abstract A graph is k-edge-choosable if for any assignment of a list of at least k colours to each edge, there is a proper edge-colouring of the graph such that each edge is assigned a colour from its list. Any loopless cubic graph G is known to be 4-edge-choosable by an extension of Brooks Theorem. In this thesis, we give an alternative proof by relating edge-choosability to the coefficients of a certain polynomial using Alon and Tarsi s Combinatorial Nullstellensatz. We interpret these coefficients combinatorially to show that the required edge-colourings exist. Moreover, we show that if G is planar with c cut edges, then all but c of the edges of G can be assigned lists of at most colours. iii

4 He gives wisdom to the wise and knowledge to those who understand. Daniel : iv

5 Acknowledgments I would like to thank Dr. Luis Goddyn, my senior supervisor, for his support, guidance and patience, as well as for suggesting this problem. I would also like to thank Dr. Ladislav Stacho, my co-supervisor, for his support and especially for his help in bringing me to Simon Fraser University. I am grateful to all the members of my examining committee for carefully reading this thesis and sharing their insights. Thanks also to all my friends and colleagues at SFU for making my time here both enjoyable and enlightening. My abundant gratitude goes to my family for their constant support, love, and prayers. And my thanks go to Kael, not least of all for keeping me sane. I would like to acknowledge the contributions made by Dr. Goddyn and Dr. Sabin Cautis (now of Columbia University), who began work on this problem during an undergraduate research project of the latter. I acknowledge the financial support of the Natural Sciences and Engineering Research Council of Canada (NSERC) and of Simon Fraser University. I also acknowledge the IRMACS Centre for hosting me during my studies. v

6 Contents Approval Abstract Quotation Acknowledgments Contents List of Figures ii iii iv v vi viii Introduction The Polynomial Method and Star Labellings 4 Cubic Graphs 8. -Edge-Connected Cubic Graphs Hamiltonian -Factor Factor Containing Multiple Cycles General Cubic Graphs Structure of the Graph Edge Weighting of the Graph Star Labellings of the Graph Proof of Theorem vi

7 4 Cubic Planar Graphs Edge-Connected Cubic Planar Graphs General Cubic Planar Graphs Structure of the Graph Edge Weightings of the Graph Reference Star Labelling of the Graph Arbitrary Star Labellings of the Graph Signs of the Star Labellings of the Graph Proof of Theorem Bibliography 46 vii

8 List of Figures. A graph G and its line graph L(G) A graph with two star labellings consistent with the edge weighting W The edge weighting W G of a graph G with a single cycle in its -factor One of the two star labellings that are consistent with W G,F of a cubic graph G with a single cycle in its -factor The edge weights assigned to an unpaired cycle and its connecting edges....4 The preliminary labels for an unpaired cycle The unique star labelling for an unpaired cycle The edge weights assigned to a paired cycle and its connecting edges The unique star labelling for a paired cycle An example of a graph G with its tree T and graphs H Comp(G \ B) with the corresponding H The edge weights for H Comp(G \ B) of Type() and Type() The edge weights for threads of H Comp(G \ B) of Type () The subgraph C b of G in Lemma The unique star labelling for H Comp(G \ B) of Type () or Type ()..... The unique star labellings for threads The primary and secondary edge weights for the edges of threads of H Comp(G \ B) of Type () The edge weighting W of the example graph from Figure The star labelling ρ for H Comp(G \ B) The reference star labelling ρ for the example graph from Figure viii

9 4.5 The general star labelling π for a blue thread of H Comp(G \ B) of Type() The general star labelling π for red or green threads of H Comp(G \ B) of Type () Applying secondary even (option ) case labels to a thread of length m leads to a contradiction ix

10 Chapter Introduction We consider a graph G = (V, E), possibly having multiple edges, but with no loops. Basic graph theory definitions can be found in West s Introduction to Graph Theory [5]. Given a set of colours C, a list assignment for the edges of G is L = {L e } e E, where L e C. For a list assignment L, a L-edge-colouring of G is a proper edge-colouring c : E C of G where c(e) L e. Similarly, we can define a list assignment for vertices, L = {L v } v V, and a proper colouring of the vertices that uses colours from these lists is an L-colouring. We say G is k-edge-choosable for a vector k = (k e ) e E N E if G is L-edge-choosable, for every list assignment L such that L e k e for each edge e E. The smallest integer k such that G has a k-edge-colouring is the chromatic index, χ (G) = k. The smallest integer k such that G is (k,..., k)-edge-choosable is the list chromatic index, χ l (G) = k. It is obvious that χ (G) χ l (G), since the edge list-colouring where each edge is assigned the list {,,... χ l (G)} gives a proper χ (G)-edge-colouring of G. In fact, the Listcolouring Conjecture states that χ (G) = χ l (G) for any graph G. Jensen and Toft s Graph Colouring Problems [] cites Häggkvist and Chetwynd [8], who assert that several people thought of this conjecture independently, including V.G. Vizing, R.P. Gupta, and M.O. Albertson and K.L. Collins but that it was first published by Bollobás and Harris in 985 [4]. This is a difficult conjecture to prove; it has been verified for several classes of graphs including, among others, bipartite graphs [7], -factorable planar graphs [6], and K n where n is odd [9]. The last two of these results were proven using the polynomial method, which is described in Section. In our case, the polynomial of interest for the polynomial method is the graph monomial (we are following the terminology of [6]).

11 CHAPTER. INTRODUCTION G L(G) Figure.: A graph G and its line graph L(G). Definition Let x = (x v ) v V. The graph monomial of G = (V, E), p G (x), is p G (x) = (x u x v ) µ(u,v) uv E where µ(u, v) is the number of edges of E joining u, v V. Let L = {L v } v V be a list assignment with L v Z for each vertex. Then, we see that G has an L-colouring if and only if there exist s v L v for each vertex v V, such that p G ((s v ) v V ). We define the line graph, L(G), of G = (V, E). The set of vertices of L(G) is the set of edges of G, E. Two vertices of L(G), e, e E, are connected by the same number of edges in L(G) as they have common endpoints in G. For example, two parallel edges of G become two vertices of L(G) joined by parallel edges. Thus, list edge-colourings of G correspond to list colourings of L(G). Combining the graph monomial with the line graph gives us an algebraic test for the existence of a list edge-colouring of G: for a list assignment L = {L e } e V, G has a L-edge-colouring if and only if there exist s e L e for each edge e E such that p L(G) ((s e ) e E ). A Brooks type theorem combines with the line graph to give us our first result about edge-choosability. Theorem.. (Brooks Theorem for Choosability [4]) Every loopless connected graph G that is neither a complete graph nor an odd cycle is (G)-choosable.

12 CHAPTER. INTRODUCTION Since the line graph of a cubic graph is 4-regular but is not the complete graph on 5 vertices, the following corollary is immediate. Corollary.. If G is a cubic graph, then χ l (G) 4 Later, we will see a result of Ellingham and Goddyn [6] that implies that if χ (G) = for a planar cubic graph G, then χ (G) = = χ l (G). So Corollary.. implies that the list-colouring conjecture is true for planar cubic graphs. In this thesis, we consider the edge-choosability cubic graphs G using the polynomial method. From Corollary.., we already know that χ l (G) 4. We will reprove this result and give tighter bounds on the list sizes, which are particularly strong in the case where G is also planar. In Chapter, we introduce the Combinatorial Nullstellensatz, the polynomial method and star labellings of a graph, and we show how they relate to list edge-colourings of a graph. Chapter and Chapter 4 each prove one the following theorems. Theorem.. If G is a cubic graph, then χ l (G) 4. Note that Theorem.. is the same as Corollary... Not only do we use a different proof method, but our proof shows that about E(G) edges use smaller list size, and it also gives a stronger result about the coefficients of the line graph monomial of G (see Section ). When G is planar, we get a stronger result. Theorem..4 If G is a planar cubic graph with c cut edges, then G is k-edge-choosable for some k = (k e ) e E(G) (4, 4..., 4), where k e = 4 holds for at most c edges.

13 Chapter The Polynomial Method and Star Labellings The polynomial method is based on the Combinatorial Nullstellensatz; a proof of the following theorem can be found in []. Theorem..5 (Combinatorial Nullstellensatz) Let F be a field, and let f = f(x, x,..., x n ) be a polynomial in F[x, x,..., x n ]. Suppose that deg(f) = where each t i is a nonnegative integer and that the coefficient of n i= x t i i n t i, i= in f is nonzero. If S, S,..., S n are subsets of F with S i > t i, then there exist s S,... s n S n so that f(s,..., s n ). In general, the polynomial method is the application of the Combinatorial Nullstellensatz to a given polynomial to prove list related properties of objects such as graphs, hypergraphs and groups. Alon and Tarsi pioneered this method in [] and applications of this method to many different areas are collected in []. Given a graph G, we will apply the polynomial method to find a bound on the list edge-choosability of G by interpreting the coefficients of p L(G) (x) combinatorially using star labellings of the graph. The following definitions are based on [6]. A star labelling π v of a vertex v V is a bijection π v : δ(v) {,,..., deg(v) }, where δ(v) is the set of edges incident with v. If π v (e) = k, then we say v is incident to the label k at the edge e. A star labelling π of G is a function that assigns a star labelling 4

14 CHAPTER. THE POLYNOMIAL METHOD AND STAR LABELLINGS Figure.: A graph with two star labellings (centre and right) consistent with the edge weighting W (left). to each vertex of G, that is π : v π v. For a given graph, we fix a star labelling ρ to be the reference labelling of the graph. We let S [n] be the permutation group of the set {,..., n }. For every vertex, we define the sign of a star labelling of a vertex to be sgn(π v ) = sgn(π v ρ v ), since π v ρ v S [deg(v)]. Then, the sign of a star labelling of the graph G is sgn(π) = sgn(π v ). v G Star labellings of a graph G are closely associated with edge weightings, W : E Z E. A star labelling is consistent with an edge weighting W if for all e = uv E, π u (e) + π v (e) = W (e). Figure. shows a graph G with an edge weighting W (left) and two star labellings of G consistent with W (centre and right). If we take the centre star labelling to be the reference labelling ρ, then the right labelling has negative sign, since the labelling of each of the three vertices incident with the outer face differs from ρ by one transposition, and the star labelling of the fourth vertex is the same. The graph monomial of a graph G is of uniform degree, so if any term α of p L(G) (x) has nonzero coefficient, then G is (t e + ) e E -edge-choosable. e E(G) x te e We prove the following theorem by relating the coefficients of the graph monomial of L(G) to the star labellings of G.

15 CHAPTER. THE POLYNOMIAL METHOD AND STAR LABELLINGS 6 Theorem..6 Consider a graph G = (V, E) with an edge weighting W. If the number of star labellings of G of positive sign consistent with W is not equal to the number of star labellings of negative sign consistent with W, then G is (W + )-edge-choosable, where (W + ) = (W (e) + ) e E. Proof Suppose an edge weighting W satisfies the hypothesis. We will show that the coefficient of e E x W (e) e in p L (G)(x) is ± {sgn(π) : π is a star labelling of G consistent with W }. Then our hypothesis tells us that this coefficient is non-zero, and so the result holds by the Combinatorial Nullstellensatz. We consider the graph monomial of the line graph of G, and we fix a reference star labelling ρ of G. For u V and i deg(v), we define e (u,i) = ρ u (i) = e E such that ρ u (e ) = i. We denote the number of common end points of e, e E by ν(e, e ). p L(G) (x) = {(x e x e ) ν(e,e ) : e e ; e, e E} = ± ( {(xe x e ) : e e, e and e E are incident with u}) u V = ± ( ) (x e(u,i) x e(u,j) ). u V i<j For each u V the product is a Vandermonde determinant, p L(G) (x) = ± det({x i e (u,j) } i,j,...deg(u) ) u V = ± sgn(σ)x σ() e (u,) x σ() e (u,) x σ(deg(u) ) σ S [deg(u)] u V (G) e (u,deg(u) ).

16 CHAPTER. THE POLYNOMIAL METHOD AND STAR LABELLINGS 7 The set of permutations S [deg(u)] is {π u ρ u If π u ρ u = σ S [deg(u)], then sgn(π u ) = sgn(σ). Thus, p L(G) (x) = ± sgn(π u )x πu(e(u,)) e (u,) u V (G) star labellings π u = ± = ± star labellings π of G W Z E sgn(π) e=uv E : π u is a star labelling of u}. x πu(e (u,)) e (u,) x πu(e)+πv(e) e x πu(e (u,deg(u) )) e (u,deg(u) ) ( ( ) {sgn(π) : π is a star labelling consistent with W } e E x W (e) e So to show edge-choosability for a graph G, we can find an edge weighting W that is consistent with unequal numbers of positive and negative star labellings. One easy way to do this is to choose an edge weighting where all consistent star labellings have the same sign. This is the approach used in the proof of Theorem... A second approach, used in the proof of Theorem..4, is to consider multiple edge weightings of G. Corollary..7 Consider a graph G = (V, E) and a set W of edge weightings of G. If the number of star labellings of G of positive sign consistent with any W W is not equal to the number of star labellings of negative sign consistent with any W W, then G is (W + )-edge-choosable for at least one W W. Also, G is (W max + )-edge-choosable, where W max (e) = max{w (e) : W W}. Proof For at least one W W the number of star labellings of G of positive sign consistent with W is not equal to the number of star labellings of negative sign consistent W. By Theorem..6, G is (W + )-edge-choosable and W W max, so G is (W max + )-edge-choosable. ).

17 Chapter Cubic Graphs In this chapter, we consider any connected cubic graph G and prove Theorem.... -Edge-Connected Cubic Graphs We first consider the case where G is -edge-connected. Then G has -factor [5, p.9]. Let F be a -factor of G with k component cycles. We denote the set of cycles of F by F. In Section.. and Section.., we will use this -factor F to define an edge weighting W G,F of G and explore the star labellings that are consistent with W G,F... Hamiltonian -Factor First, we consider the case where the -factor F consists of a Hamiltonian cycle, so k =. Then the graph G is an even cycle C = v e v... v m e m v of length m with m define the edge weighting W G,F by, W G,F for e = e i E(C), i =... m (e) = for e = c, a chord of G as illustrated in Figure.. chords. We (.) Lemma.. A -edge-connected cubic graph G with a -factor consisting of a single cycle has exactly two star labellings consistent with W G,F labellings have the same sign. as defined in (.), and these star 8

18 CHAPTER. CUBIC GRAPHS 9 v e v e m v m Figure.: The edge weighting W G of a graph G with a single cycle in its -factor. v v v m Figure.: One of the two star labellings that are consistent with W G,F of a cubic graph G with a single cycle in its -factor.

19 CHAPTER. CUBIC GRAPHS Proof Any star labelling π consistent with W G,F must have the labels π vi (c) = at each v i V (C) where c is the chord with endpoint v i. Since π is a star labelling, one of the labels π vi (e i ) and π vi (e i ) is and the other label is. One possible star labelling is seen in Figure.. Since the star labelling π is consistent with W G,F, π is one of two possible star labellings, π and π, which are defined for each vertex v i for i =,... m by, for e = c, the chord with endpoint v i (π ) vi (e) = for e = e i for e = e i for e = c, the chord with endpoint v i (π ) vi (e) = for e = e i for e = e i Since these labellings differ by a transposition at each vertex, sgn((π ) vi ) = sgn((π ) vi ) for i =,..., m and so sgn(π ) = ((π ) v ) = ((π ) v ) = sgn(π ) v V (G) v V (G) since V (G) is even... -Factor Containing Multiple Cycles Suppose instead that the -factor F has k cycles. We say two cycles C, C F, are adjacent if there is a connecting edge of C, uv E(G), with u V (C ) and v V (C ). We also say uv is a connecting edge of u and v. We fix a maximal pairing of adjacent cycles. We let P F be the set of paired cycles of F and U = F \ P be the set of unpaired cycles. We note that because the pairing is maximal, each unpaired cycle C U is adjacent only to cycles in P. For each cycle C P, we designate a connecting edge between C and its pair C as the link edge of both C and C, c C = c C. For a cycle C U, we pick any connecting edge of C to be its link edge, c C. For each C = v e... v m e m F, we may assume that v V (C) is the endpoint of c C. Then v is the base vertex of C and we designate the edge e as the base edge of C.

20 CHAPTER. CUBIC GRAPHS We can now define the edge weighting W G,F by, for e = e is a base edge in some C F for e = c C is a link edge for some C P W G,F (e) = for e = c C is a link edge for some C U (.) for e = e i is a non-base edge in some C F for all other edges e of G The following star labelling of G, ρ, is consistent with W G,F. The star labelling ρ is illustrated in Figure.5 and Figure.7. For each cycle C = v e... e m v F for i =, j = for i =, j = ρ vi (e j ) = for i = j for i, i = j + mod m (.) ρ v (c C ) = (.4) ρ vi (c) = for all cord edges and connecting edges c c C with endpoint v i (.5) The two following lemmas, Lemma.. and Lemma.., show that ρ is the unique star labelling of G that is consistent with W G,F. In the former, we consider the star labellings of the vertices of the unpaired cycles, and in the latter, we consider the star labellings of the vertices of the paired cycles. Lemma.. Let G be a -edge-connected cubic graph with a -factor F of G of at least cycles. If the star labelling π of G is consistent with W G,F as defined in (.), then we have π v = ρ v for every vertex v of each C U. In particular, the label of the base vertex at the link edge is π v (c C ) =. Proof We recall that W G,F assigns edge weights as shown in Figure.. Each edge c that is a chord or connecting edge of F with endpoint v i for i (so c is not c C ), has edge weight W (c) =. So the label must be π vi (c) = = ρ vi (c). For the link edge of C, e C, we recall that W G,F (c C ) =, and we let δ = π v (c C ) so δ {, }, as shown in Figure.4.

21 CHAPTER. CUBIC GRAPHS v e v e m c C v m Figure.: The edge weights assigned to an unpaired cycle and its connecting edges. The base and link edges are in bold. v e δ v e m v m c C Figure.4: The preliminary labels for an unpaired cycle.

22 CHAPTER. CUBIC GRAPHS v e v e m v m c C Figure.5: The unique star labelling for an unpaired cycle. For all non-base cycle edges, e i E(C) i, the edge weight is W G,F (e i ) =, and the edge weight of the base edge e is W G,F (e ) =. Since all vertices are incident to the labels, and, summing the labels over all vertices of the cycle we have, W G,F (e) + δ = ( + + ) V (C) e E(C) δ = V (C), and so π v (c C ) = δ = = ρ v (c C ). Since e n has edge weight W G,F (e n ) =, the other labels of v are π v (e n ) = = ρ v (e n ) and π v (e ) = = ρ v (e ). These labels are shown as the boxed labels in Figure.5. Then, as the base edge e has edge weight W G,F (e ) =, we know π v (e ) = = ρ v (e ). We know v is incident with the label at its chord or connecting edge, so the final label incident with v is π v (e ) = = ρ v (e ). Similarly, we see that the remaining labels for the vertices in V (C) are π vi (e i ) = = ρ vi (e i ) and π vi (e i ) = = ρ vi (e i ) for i =,..., n, as shown in Figure.5. So π v = ρ v for all v V (C).

23 CHAPTER. CUBIC GRAPHS 4 v e v e m v m c C Figure.6: The edge weights assigned to a paired cycle and its connecting edges. The link and base edges are in bold. Lemma.. Let G be a -edge-connected cubic graph with a -factor F of G of at least cycles. If the star labelling π of G is consistent with W G,F as in (.), then we have π v = ρ v for every vertex v of each C P. In particular, the label of the base vertex at the link edge is π v (c C ) =. Proof We recall that the edge weighting W G,F is as shown in Figure.6. We consider any connecting edge c of C with endpoint v i V (C) that is not the link edge c C. The connecting edge c may be a link edge of some unpaired cycle and so have edge weight W G,F (c) =, or else c has edge weight W G,F G(c) =. In both cases the labelling at v i must have π vi (c) = = ρ vi (c). In the second case, this label follows since the endpoint u of c in the unpaired cycle must have the label π u (c) = by Lemma... By the same double counting argument as in the proof of Lemma.., we see that any star labelling of V (C) consistent with W G,F must have the label π v (c C ) = = ρ v (c C ). Again, as in the proof of Lemma.., since e n has edge weight W G,F (e n ) =, the other labels of v are π v (e n ) = = ρ v (e n ) and π v (e ) = = ρ v (e ), these labels are shown as

24 CHAPTER. CUBIC GRAPHS 5 v e c v C e m v m Figure.7: The unique star labelling for a paired cycle. the squared labels in Figure.7. Since the base edge e has edge weight W G,F (e ) =, we know π v (e ) = = ρ v (e ). We know v is incident with the label at its connecting edge so the final label incident with v is π v (e ) = = ρ v (e ). Similarly, we see that the remaining labels for the vertices in V (C) are π vi (e i ) = = ρ vi (e i ) and π vi (e i ) = = ρ vi (e i ) for i =,..., n, as seen in Figure.7. So π v = ρ v for all v V (C). Corollary..4 Let G be a -edge-connected cubic graph and F a -factor of G of at least cycles. Then there is a unique star labeling π of G consistent with W G,F as defined in (.). Proof This follows immediately from Lemma.. and Lemma... Lemma..5 Any -edge-connected cubic graph G is 4-edge-choosable. Proof Let F be a -factor of G with k cycles, and let W G,F be the corresponding edge weighting as defined in (.) if k = or in (.) if k. If F consists of a single cycle, then by Lemma.., there are exactly two star labellings of G consistent with W G,F, and both have the same sign. Otherwise, Lemma..4, there is exactly one star labelling of G

25 CHAPTER. CUBIC GRAPHS 6 consistent with W G,F. In both cases, there are a different number of star labelling with positive and negative sign. By Theorem..6, G is 4-edge-choosable since W G,F.. General Cubic Graphs Now we consider any connected cubic graph G... Structure of the Graph We let B be the set of cut edges of G, so G\B consists of -edge-connected components and isolated vertices, and we denote the set of these components by Comp(G \ B). The vertices of a subgraph H Comp(G \ B) have degrees, or, and H must have an even number of odd degree vertices. Thus, H must be of one of the following types: Type () a single vertex, Type () a cycle, or Type () a graph with at least two vertices of degree. Let T be the tree formed from G by contracting each subgraph H Comp(G \ B) to a single vertex. Then, T has edges E(T ) = B and vertices V (T ) = {H : H Comp(G \ B)}. We root T at a leaf H r. The root H r must be of Type (), since for any vertex H of T of Type () or Type (), H must be adjacent to more than one cut edge in T. For each H H r, consider the cut edge b H B that is above it in T (which is incident to H on the path from H r to H in T ). The endpoint of b H that lies in V (H) is the root vertex of H. Other edges of B that are incident with H in T (and G) are referred to as the bridge edges below H, and each such bridge edge is below the vertex v V (H) that is its endpoint. We denote the set of bridge edges below H by B(H), and the bridge edge below a vertex v by B(v). Consider an H of Type (). Any maximal path t = v e... v m e m v m+ of length m and such that all internal vertices have degree is called a thread of H. We let V in (t) = {v,... v m } be the set of internal vertices of t. Construct the graph H by suppressing all internal vertices of the threads of H, that is, by replacing every thread t with a single edge e t = v v m+. Each edge e t in H that corresponds to a thread t in H is called a thread edge. All edges of E(H) E(H ) are referred to as non-thread

26 CHAPTER. CUBIC GRAPHS 7 edges. Since the root vertex of H H r has degree, it is in a thread. We designate the corresponding thread edge as the root edge of H. Let G = {H : H Comp(G \ B), H is of Type ()}. For each H Comp(G \ B), we will distinguish certain vertices and edges of H and B(H). If H is of Type (), then H is a single vertex, which we name v, and we name the two cut edges in B(H) b and b. If H is of Type (), then we let the cycle H be v e v e... v n e n v, where v is the root vertex. We let the edges in B(H) be b i = B(v i ), for i =,..., n. If H is of Type (), then each thread t is t = v e... v m e m v m+, and we let b i = B(v i ) for i =,... m. If some v j V (t) is the root vertex of H, then b j does not exist. Let B(t) = v V (t) B(v) be the set of cut edges below the thread t... Edge Weighting of the Graph We define an edge weighting W of G by the following procedure, which assigns edge weights to the edges E(H) B(H) for each H Comp(G \ B). If H is of Type (), then we assign the edge weights W (b ) = and W (b ) =. If H is of Type (), then we assign the edge weights W (e ) = and W (b ) =. For every other edge f (E(H) B(H)) \ {e, b }, the edge weight is W (f) =. These edge weights are illustrated in Figure.9. For both Type () and Type (), the average edge weight over E(H) B(H) is. Finally, we consider an H Comp(G \ B) of Type (). This includes the case where H = H r. Each such H corresponds to some H. In the non root case, each H is -edge-connected and cubic and has a root edge e. We can find a perfect-matching of H that includes e []. The complement of this perfect matching is a -factor F H. The root edge, e, is either a chord of a cycles C F or a connecting edge between two cycles C, C F. If e is a connecting edge, then we take a maximal pairing that contains the pair C, C. If e is a chord we take any maximal pairing. For the root H r, we take any -Factor F H r of H r and any maximal pairing of cycles. Either way, we define the edge weighting W H,F H as in (.) or (.) and fix a star labelling π of H consistent with W H,F H. We assign each non-thread edge e E(H) E(H ) the weight W (e ) = W H,F H (e ). We consider each thread t = v e v... e n v m+ of H and the corresponding thread edge e t = v v m+ of H, and assign edge weights to the edges E(t) B(t) according to the procedure below which depends on the labels πv (e t ) and πv m+ (e t ). We may assume that πv (e ) πv m+ (e m ). There are five cases, denoted Case(s,t) where s = πv (e t ) and t = πv m+ (e t ), so s t. The resulting edge weights are illustrated in Figure..

27 CHAPTER. CUBIC GRAPHS 8 G H H i Type H i H i * () H 6,6, () 9, H 5 H 4 H () H 7 H 8 4 () NA H 9 5 () NA H T H 7 () H 8 () H 6 H 5 H 4 H H 7 H 8 Root vertices and edges are in bold. H H 9 Figure.8: An example of a graph G with its tree T and graphs H Comp(G \ B) with the corresponding H.

28 CHAPTER. CUBIC GRAPHS 9b (): Type (): v Type v b v n b v e Figure.9: The edge weights for H Comp(G \ B) of Type() and Type(). Case(,): We assign the edge weights W (e ) =, W (e ) = and W (b ) =. We assign the edge weight W (f) = to the remaining edges f (E(t) B(t)) \ {e, e, b }. Case(,): We assign the edge weights W (e ) = and W (e m+ ) =. We assign edge weight W (f) = to the remaining edges f (E(t) B(t)) \ {e, e m+ }. Case(,): We assign the edge weights W (e ) =, W (e ) = and W (b ) =. We assign the edge weight W (f) = to the remaining edges f (E(t) B(t)) \ {e, e, b }. Case(,): We assign the edge weights W (e m ) =, W (e m ) = and W (b m ) =. We assign edge weight W (f) = to the remaining edges f (E(t) B(t))\{e m, e m, b m }. Case(,): We assign the edge weights W (e ) =. We assign edge weight W (f) = to the remaining edges f (E(t) B(t)) \ {e }. We note that a root thread t corresponds to a root edge e t, which is either a link edge connecting paired cycles with weight W H,F H (e t ) = and labels πv (e t ) = and πv m+ (e t ) = (by Lemma..), or chord of weight W H,F H (e t ) = with labels πv (e t ) = and πv m+ (e t ) =. That is e t is either in Case(,) or Case(,) when t is a root thread. Lemma.. For any H Comp(G \ B), the average edge weight W (e) over the edges in E(H) B(H) is. Proof We have already seen this for H of Type () or Type () at the beginning of this section. For H of Type () with -factor F H as chosen above, we can see from Figure.

29 CHAPTER. CUBIC GRAPHS Case(,): v v v m+ v e t e e v m m+ b e Case(,): v v v m+ e v e t v m m+ b b m e Case(,): v v v m+ e e v e t v m m+ b e Case(,): v m+ v m v e t e m- m+ v e m b m e v Case(,): v v v m+ e e v e t m v m+ b e Figure.: The edge weights for threads of H Comp(G \ B) of Type (). Note that the middle section of each diagram is repeated or more times to make a thread of length m..

30 CHAPTER. CUBIC GRAPHS that the total edge weight for each thread t of H, with corresponding thread edge e t, is W H,F H (e t ) + E(t) + B(t). We note that the cubic graph H has a star labelling that is consistent with the edge weighting W H,F H. One such star labelling is given by π in Section.., if F H has only one cycle, or by ρ in Section.., if F H has multiple cycles. Thus we know that (e) = ( + + ) V (H ) e E(H W H,F H and by the handshake theorem, (e) = ( E(H ) ) e E(H W H,F H = E(H ). So the sum of the edge weights in E(H) B(H) is, e E(H) B(H) W (e) = e t e E(H ) W H,F H (e) + {W H,F H (e t ) + E(t) + B(t) : t is a thread of H} = W H,F H (e) + { E(t) + B(t) : t is a thread of H} e E(H ) ( = E(H ) + ) { E(t) + B(t) : t is a thread of H}. The set of edges E(H) B(H) is the set of edges E(H ) with each thread edge e t replaced by the edges E(t) B(t). Then, E(H) B(H) = E(H ) + { E(t) + B(t) : t is a thread of H}. Thus the average edge weight over E(H) B(H) is... Star Labellings of the Graph We define the lower endpoint of a cut edge b B, low(b), to be the endpoint of b farthest from H r in T ; the other (nearer) endpoint of b is the upper endpoint, up(b). We say a star labelling π is balanced if for every cut edge b, π low(b) (b) =.

31 CHAPTER. CUBIC GRAPHS b α C Figure.: The subgraph C b of G in Lemma... Lemma.. Any star labelling π of G that is consistent with the edge weighting W defined in Section.. is balanced. Proof Consider any cut edge b B and the component C b of G \ b that contains low(b). By Lemma.., the average edge weight over E(H) B(H) for each H G \ B is. Thus, the average edge weight over E(C b ) is. Let π low(b) (b) = α be the lower label of b as in Figure.. The sum of labels around a vertex is + + =, and the sum of labels for any edge is its weight. So summing the labels of π in G around every vertex in V (C b ) gives, V (C b ) = α + W S (e) = α + E(C b ). e E(C b ) Now, V (C b ) is odd since C b must have an even number of degree vertices and one vertex of degree. So α {,, } must be odd, that is α =. Thus π is balanced. As an immediate consequence of Lemma.., we see that if b B is a cut edge of G, then any star labelling π of G that is consistent with W must have the labels π low(b) (b) = and π up(b) (b) = W (b). Lemma.. All star labellings of G that are consistent with W defined in Section.. have the same sign. Proof We consider each H Comp(G \ B) in turn and show that the overall sign in V (H), π v, is the same for all possible star labellings π of G consistent with W. v V (H)

32 CHAPTER. CUBIC GRAPHS Type (): Type (): v b b v e b (a) (b) Figure.: The unique star labelling for H Comp(G \ B) of Type () or Type (). If H is of Type (), then the labelling at v must be π v (b H ) =, π v (b ) = and π v (b ) =, since π is balanced. This labelling is shown in Figure. (a). So all star labellings consistent with W have the same labels, and thus the same sign, for V (H) = {v }. For H of Type (), since π is balanced, the labelling must have labels π v (b H ) =, π v (b ) = and π vi (b i ) = for i =,..., n. The edge weight of e is W (e ) = and π v (b H ) =, so π must have π v (e ) = and π v (e ) =. We must have π v (e n ) =, as π v (b H ) = and π v (e ) =. The edges e i for i =,... n have edge weight. These edges must have labels and, since π vi (b i ) = for i,. In fact, π vi (e i ) = and π v(i+) (e i ) = (indices are taken modulo n) since they must agree with the label π v (e n ) =. This labelling is shown in Figure. (b). So all star labelling consistent with W have the same labels, and thus the same overall sign, for V (H). Finally, we consider H of Type () with the -factor F H chosen in Section... First we show that for each thread t, there is a unique star labelling for the vertices in V in (t) consistent with the edge weights W. This unique star labelling is illustrated in Figure.. We distinguish certain labels, by boxing or triangling them in Figure.; these labels are important in the following argument. Let e t E(H ) be the corresponding thread edge to the thread t. We recall the edge weights W (e) of the edges e E(t) B(t) assigned in Section.. and shown in Figure..

33 CHAPTER. CUBIC GRAPHS 4 Case(,): v v v m+ v e t e e v m m+ b e Case(,): v v v m+ v e t e e v m m+ b b m e Case(,): v v v m+ v e t e e v m m+ b Case(,): v m+ v m v v e t e m- e m+ v b m Case(,): v v v m+ v e t e e v m m+ b e e m e Figure.: The unique star labellings for threads.

34 CHAPTER. CUBIC GRAPHS 5 We will show that the labels π v for v V (t) are as shown in Figure.. We suppose t is not a root thread. In each case the squared labels, π vi (b i ), are determined to be as shown by Lemma... Next, we see that the triangled label, π v (e ), π v (e ) or π vm (e m ) is as shown, based on the edge weight W (e ), W (e ) or W (e m ) and the fact that each vertex must be incident to the labels, and. Finally, we see that the rest of the labels π v (e i ) are as shown, again since π is a star labelling consistent with W. If t is a root thread, then t has edge weights W given by either Case(,) or Case(,). In Figure., we see that in both these cases all cut edges b B(t) have the label π up(b) (b) =. If v j is the root vertex of H, then π vj (b H ) =, and the remaining labels of π are as above. Thus, there is a unique star labelling that is consistent with the edge weighting W for the vertices V in (t) of the root thread of H. For each star labelling π of the vertices V (H) that is consistent with W, we can define a star labelling π of H by π v = π v for v V (H ) (where we identify π v (e t ) = π v (e ) and π v m+ (e t ) = π vm+ (e m ) for each thread t = v e... e m v m+ and corresponding thread edge e t = v v m+. The star labelling π is consistent with W H,F H since for all labellings in Figure. π v (e )+π vm+ (e m ) = W H,F H (e t ), and W (e) = W H,F H (e) for all non-thread edges e. By Lemma.. and Lemma.., sgn(π ) is the same for all choices of π. For every star labelling π consistent with W, π v is as shown in Figure. for v V (t) and every thread t of H so, sgn(π v ) = sgn(π v ) {sgn(π v ) : v V in (t) for a thread t of H} v V (H) v V (H ) = sgn(π v) {sgn(π v ) : v V in (t) for a thread t of H} v V (H ) = sgn(π ) {sgn(π v ) : v V in (t) for a thread t of H} and so the signs of all star labellings π that are consistent with W are the same...4 Proof of Theorem.. Proof of Theorem.. We can construct a star labelling π of G consistent with W as follows. We let π low(b) (b) = for every b B. For vertices of H Comp(G \ B) of Type () or Type (), we let π v be as shown in Figure.. For each H Comp(G \ B) of Type (), we choose a -factor F H as in in Section.. and we fix a star labelling π of H consistent with W H,F H ; such a star labelling is illustrated by π in Section..

35 CHAPTER. CUBIC GRAPHS 6 or by ρ in Section... Then we let π v = πv for all v V V (H ). Finally, for each for v V in (t) for every thread t of H, we let the star labelling π v be as shown in Figure., where the cases are defined as in Section.. and depend on π above. By Lemma.., all star labellings of G that are consistent with the edge weighting W, as defined in Section.., have the same sign. So the number of positive and negative star labellings must be different, and by Theorem..6, G is 4-edge-choosable since W.

36 Chapter 4 Cubic Planar Graphs In this chapter, we consider any connected cubic planar graph G and prove Theorem Edge-Connected Cubic Planar Graphs We consider a -edge-connected cubic planar graph G. By a consequence of the Four Colour Theorem [], G is -edge-colourable. Then, by [6, Theorem.] of Ellingham and Goddyn, G is -edge-choosable. Their result is actually more general; it asserts that k-edge-colourable k-regular planar graphs are k-edge-choosable. For Section 4. we need another theorem of [6, p.45 and Theorem.] regarding the star labellings of G. First, we must define the following terminology. We let be the edge weighting of a graph that assigns each edge weight. For a star labelling π of any graph, we consider the subgraph with edges {e = uv : π u (e) =, π v (e) = }. The cycles of this subgraph are called - cycles of the graph, and π is - bipartite if all the - cycles have even length. Theorem 4.. If G is a -edge-connected cubic planar graph, then all - bipartite star labellings π of G consistent with have the same sign. 7

37 CHAPTER 4. CUBIC PLANAR GRAPHS 8 4. General Cubic Planar Graphs 4.. Structure of the Graph We consider a -edge-connected cubic planar graph G. Many of the definitions in this section are the same as in Section.., but we repeat some here for clarity. In particular, we let B E be the set of cut edges of G, so G\B consists of -edge-connected components and isolated vertices. As in Section.., each H Comp(G\B) must be of one of the following types: Type () a single vertex, Type () a cycle, or Type () a graph with at least two vertices of degree. Again, we let T be the tree with vertices H Comp(G \ B) and edges B, and we root T at a leaf H r. For each H Comp(G \ B), the root vertex and the bridge edges below H are as before. For H is of Type (), we recall the definition of a thread t = v e... v m e m v m+ of H. Again, we construct the graphs H with corresponding thread edges e t = v v m+, including the root thread edge. We recall the definition of G = {H : H Comp(G \ B), H is of Type ()}. As in Section.., we will distinguish certain vertices and edges of H Comp(G \ B) and B(H). Recall that for H of Type (), H is the vertex v, and the two cut edges in B(H) are b and b. If H is of Type () with root vertex v, H is the cycle v e v e... v n e n v, and the edges in B(H) are b i = B(v i ) for i =,..., n. If H is of Type (), then each thread t of H is t = v e... v m e m v m+ and we let b i = B(v i ) for i =,... m. If v j V (t) is the root vertex of H then b j is not defined, rather v j is adjacent to the cut edge b H above it. Every component H of G is -regular, planar and -edge-connected, and thus -edgecolourable by the Four Colour Theorem. We fix a -edge-colouring φ {red, green, blue} of G such that: (A) each root edge is coloured blue, and (B) the number of blue thread edges is maximal subject to (A). (4.) Such a edge-colouring exists since each root edge is in a distinct component of G.

38 CHAPTER 4. CUBIC PLANAR GRAPHS Edge Weightings of the Graph The following procedure assigns a primary edge weight to each edge in E, and an additional secondary edge weight to each edge of certain threads of G. For an edge e, we denote the primary edge weight by w p (e) and, if it exists, we denote the secondary edge weight by w s (e). Then we define a set of edge weightings W of G; an edge weighting will assign either the primary or secondary edge weight at each edge. We consider each H Comp(G \ B) in turn and assign a primary weight, and possibly a secondary weight, to each edge in E(H) B(H). If H is of Type () or Type (), then only a primary weight is assigned to an e E(H) B(H). If H is of Type (), then we assign a primary edge weight to each edge e E(H) B(H), and we may also assign a secondary edge weight to a subset of these edges. The edge weights for H of Type () and Type () are as in Section... In particular, if H is of Type (), then the primary edge weights are w p (b ) = and w p (b ) =. Again as in Section.., if H is of Type (), then the primary edge weights are w p (e ) =, w p (b ) =, and w p (f) = for f (E(H) B(H)) \ {e, b }. These edge weights are shown in Figure.9. Finally, we consider H of Type (). First, we define a partial orientation on the corresponding H. The red and green edges of H under φ induce cycles, which we orient clockwise. We assign only the primary edge weight w p (e) = for each non-thread edge e of H. We consider each thread t of H separately. For each thread, we assign edge weights to the edges in E(t) B(t). Let e t be the thread edge of H corresponding to t. We proceed as follows. In the case where φ(e t ) = blue, we assign only a primary edge weight to each edge in E(t) B(t). Note the e t may be the root edge of H in this case. These weights are w p (e ) =, w p (e m ) = and w p (f) = for every other edge f (E(t) B(t)) \ {e, e m }. In the case where φ(e t ) {red, green}, we assign both a primary edge weight and a secondary edge weight for each edge in E(t) B(t). Note that e t = v v m+ cannot be the root edge of Hi and that e t has an orientation; we may assume the orientation is from v to v m+. We assign the primary edge weights of w p (e m ) =, w p (e m ) =, w p (b m ) =, and w p (f) = for f (E(t) B(t)) \ {e m, e m, b m }. The secondary edge weights depend on the parity of m. If m is odd, then we assign the secondary weights w s (e ) =, w s (e ) =, w s (e ) =, and w s (f) = for f (E(t) B(t)) \ {e, e, b }. Finally, if m is even,

39 CHAPTER 4. CUBIC PLANAR GRAPHS Type (): blue: primary: v m+ v e m e v red/green: primary: v v v m e e m b b m v m+ secondary odd: v v v m+ e e m b secondary even: v v m v m+ e em b m Figure 4.: The primary and secondary edge weights for the edges of threads of H Comp(G \ B) of Type (). then we assign secondary edge weights w s (e m ) =, w s (b m ) =,and w s (f) = for f (E(t) B(t)) \ {e m, b m }. The primary and secondary edge weights are illustrated in Figure 4.. We now define the edge weightings of the graph G. Let R be the set of threads t of G such that φ(e t ) {red, green}. The set of edge weightings W is defined to be W = {W S : S R}, where for each S R the edge weighting W S {,,, } E(G) is w s (e) if e t S (E(t) B(t)) W S (e) = w p (e) otherwise. We note that all edge weightings W S of G have average edge weight over E(H) B(H) for H Comp(G \ B).

40 CHAPTER 4. CUBIC PLANAR GRAPHS H 6 H H 5 H 4 H H H 7 H H 9 H 8 Figure 4.: The edge weighting W of the example graph from Figure.8.

41 CHAPTER 4. CUBIC PLANAR GRAPHS 4.. Reference Star Labelling of the Graph In this section, we define a reference star labelling of G, ρ, that is balanced, - bipartite, and consistent with the edge weighting W. Recall that a star labelling π is balanced if for every cut edge b, π low(b) (b) =. The labels of ρ are assigned by considering the vertices V (H) for each H Comp(G \ B) in turn. Since ρ is balanced, we know that ρ l(b) (b) = for all cut edges b B. For H Comp(G \ B) of Type (), the labelling is ρ v (b ) = and ρ v (b ) =. Since ρ is balanced, ρ is consistent with W on δ(v), and v is incident to the distinct labels,, and. The star labelling in shown Figure 4.. For H Comp(G \ B) of Type(), the labelling is ρ v (e ) =, ρ v (e n ) =, ρ v (e ) =, ρ v (b ) =, ρ v (e ) =, and for i =,..., n, ρ vi (e i ) =, ρ vi (e i ) = and ρ vi (b i ) =. By construction, ρ is consistent with W for the edges in E(H), and ρ is consistent with W for the edges in B(H), since ρ is balanced. Again, each vertex is incident to distinct labels, and, and the star labelling in shown Figure 4.. To help define π for H of Type (), we define a star labelling of G, ρ, based on φ and the partial orientation from Section 4.. so that ρ is consistent with the edge weighting of G. For every blue edge e = uv of G, the labelling is ρ u(e) = and ρ v(e) =. Every red or green edge e = uv has an orientation, say from u to v. Based on the orientation, we define the labels ρ u(e) = and ρ v(e) =. Since φ is a proper -edge-colouring, every vertex in V (G ) is incident to the labels, and, and ρ is - bipartite. Finally, we use ρ to define ρ for H Comp(G \ B) of Type (). For v V (H ), we assign labels ρ v = ρ v (for each thread t = v e... e m v m+ and corresponding thread edge e t = v v m+, we identify π v (e t ) = π v (e ) and π v m+ (e t ) = π vm+ (e m )). The vertices in V (H)\V (H ) are the internal vertices of the threads of H, {V in (t) : t a thread of H}. For each thread t = v e... e m v m+ of H, the vertices V in (t) are labelled based on the colour φ(e t ) as follows: In the case where φ(e t ) = blue, the labels ρ v (e ) = and ρ vm+ (e m ) = are assigned above. The internal vertices have the labels ρ v (e ) =, ρ vm (e m ) =, and for i =,... m, ρ vi (e i ) = and ρ vi+ (e i ) =. The labelling at each b i B(t) is ρ vi (b i ) =. We recall that the edge weights are W (e ) =, W (e m ) =, and for i =,... m, W (e i ) = and W (b i ) =. As illustrated in Figure 4., the star labellings {ρ v : v V in (t)} are consistent with W for the edges E(H) B(H).

42 CHAPTER 4. CUBIC PLANAR GRAPHS v b b (): Type (): v Type v b e red/green: blue: Type (): v m+ v v v m+ e m e e em b m Figure 4.: The star labelling ρ for H Comp(G \ B).

43 CHAPTER 4. CUBIC PLANAR GRAPHS 4 In the case where φ(e t ) {red, green}, the labels ρ v (e ) = and ρ vm+ (e m ) = are assigned above. The reference labelling is ρ vm (e m ) =, ρ vm (e m ) =, ρ vm (e m )=, ρ vm (b m ) =, ρ vi (b i ) = for i m, and for i =,..., m, ρ vi (e i ) = and ρ vi (e i ) =. We recall that the edge weights are W (e m ) =, W (e m ) = and W (b m ) =, and all other edges in E(t) B(t) have edge weight. As illustrated in Figure 4., the star labellings {ρ v : v V in (t)} are consistent with W for the edges E(H) B(H). We have shown that the reference labelling ρ is valid star labelling at every vertex of G. By construction and since ρ is balanced, this reference labelling is consistent with W. We can see that every - cycle in G under ρ is a - cycles of G under ρ, and so each has even length and thus ρ is - bipartite. We have shown Lemma 4... Lemma 4.. The reference star labelling for a planar cubic graph G, ρ, as defined above is consistent with W and is - bipartite. The reference star labelling ρ of G, as above, is consistent with and - bipartite Arbitrary Star Labellings of the Graph In this section, we consider any arbitrary star labelling π of G that is consistent with any of the edge weightings W S W. We let the set of all such star labellings be Π = {π : π is a star labelling of G consistent with some W S W}. Lemma 4.. Every star labelling in Π is balanced. Proof The proof is the same as that of Lemma.. in Chapter, recalling from the end of Section 4.. that the average edge weight over E(H) B(H) is for all H Comp(G \ B). The next lemma shows that if π is a star labelling consistent with some W S W, then for certain v V (G), π v is one of only several possible labellings. Lemma 4.. We consider W S W and π Π that is consistent with W S. Let H Comp(G \ B). If H is of Type () or Type (), then the star labelling must be π v = ρ v for every vertex v V (H). If H is of Type (), then for each thread t = v... v m+ S of H and vertex v V in (t), the star labelling must be π v = ρ v. For each thread t = v... v m+ S of H if m >, then

44 CHAPTER 4. CUBIC PLANAR GRAPHS 5 H 4 H 6 H 7 H H H 5 H 8 H H 9 H Figure 4.4: The reference star labelling ρ for the example graph from Figure.8.

45 CHAPTER 4. CUBIC PLANAR GRAPHS 6 the star labelling π vi for i =,..., m is as illustrated in Figure 4.6 in the secondary odd case or secondary even (option ) case, depending on the parity of m. If m =, then the star labelling π vi for i =,..., m is as illustrated in Figure 4.6 in either the secondary even (option ) case or secondary even (option ) case. Proof We consider some π Π which is consistent with an edge weighting W S W. It follows from Lemma 4.. that π low(b) (b) = and π up(b) (b) = W S (b) = ρ up(b) (b) for every cut edge b B. First, we consider H Comp(G \ B) of Type (), so V (H) = {v } and δ(v ) = {b H, b, b } B. Since all edge weightings in W assign the same edge weights to {b, b }, π v = ρ v by Lemma 4... Next, we consider π on the vertices of H Comp(G \ B) of Type (). Again, all edge weightings in W assign the same edge weights for E(H) B(H). By Lemma 4.., we know that π v (b ) = ρ v (b ) =, and π vi (b i ) = ρ vi (b i ) = for i =,..., n. Since W S (e ) =, we must have {π v (e ), π v (e )} = {, }. Because π is balanced, we know the label π v (b H ) =, where b H is the cut edge above v. Thus, the labels must be π v (e ) =, π v (e ) = and π v (e n ) =, since v must be adjacent to the labels, and. The edges e i, for i =,... n, have edge weight W s (e i ) = and must have labels {π vi (e i ), π vi+ (e i )} = {, }, since π vi (b i ) = for i, (indices are taken modulo n). These labels must be the same for each i and π v (e n ) =, so the labels are π vi (e i ) = and π vi+ (e i ) =. Thus we see that π u = ρ u for u V (H). Finally, we consider π for the vertices of threads of H Comp(G \ B) of Type (). We let t be a thread and e t be the corresponding thread edge of H. If φ(e t ) = blue and e t is not the root thread edge, then all edge weighting W S have the same edge weights for E(t) B(t), as illustrated in Figure 4.. Since π is balanced, we must have the labels π vi (b i ) = ρ vi (b i ) for i =,... m, which are shown as the boxed labels in Figure 4.5. Next, since W (e m ) =, the triangled label must be π vm (e m ) = = ρ vm (e m ). By W S and since every vertex must be incident to labels, and, the rest of labels are as shown in Figure 4.5. In the case where e t is the root thread edge, the root vertex v j V (t) is incident with b H and B(v j ) does not exist. Since π is balanced, we still have π vj (b H ) = and so the labels in V (t) are as in the non-root case. So for all threads with φ(e t ) = blue, the star labelling is π vi = ρ vi for all vertices v i V (t). Now we consider the case where φ(e t ) {red, green}. We will show that the star labellings are as shown in Figure 4.6. In the figure, we again distinguish certain labels, by