Acyclic Edge Colouring of 2-degenerate Graphs

Transcription

1 Acyclic Edge Colouring of 2-degenerate Graphs Chandran Sunil L., Manu B., Muthu R., Narayanan N., Subramanian C. R. Abstract An acyclic edge colouring of a graph is a proper edge colouring such that there are no bichromatic cycles. The acyclic chromatic index of a graph is the minimum number k such that there is an acyclic edge colouring using k colours and it is denoted by a (G). A graph is called 2-degenerate if each of its subgraphs has a vertex of degree at most 2. The class of 2-degenerate graphs contains important special classes such as series-parallel graphs, partial 2-trees and outerplanar graphs. It was conjectured by Alon, Sudakov and Zaks that a (G) + 2, where = (G) denotes the maximum degree of the graph. We prove the conjecture for 2-degenerate graphs, and improve the bound to + 1 for series-parallel graphs. We also prove that this improved bound holds for the class of partial 2-trees. These latter results actually give best possible upper bounds. All these bounds are proved constructively, leading to effecient algorithms to produce colourings with the stated number of colours. 1 Introduction All graphs considered here are simple and finite. A proper edge colouring of G = (V, E) is a map C : E S (where S is the set of available colours) with C(e) C(f) for any pair of incident edges e, f. The minimum number of colours needed to properly colour the the edges of G, is called the chromatic index of the graph, denoted by χ (G). A proper edge colouring C is called acyclic if there are no bichromatic cycles in the graph. In other words, the union of any two colour classes induces a set of paths (i.e., linear forest) in G. The acyclic edge chromatic number (also called acyclic chromatic index), denoted by a (G), is the least number of colours required to acyclically edge colour G. The concept of acyclic colouring of a graph was introduced by Grünbaum [5]. The acyclic chromatic index and its vertex analogue can be used to bound other parameters like oriented chromatic number and star chromatic number of a graph G, both of which have many practical applications such as in wavelength routing in optical networks ([4], [12]). Let = (G) denote the maximum degree of a graph G. By Vizing s theorem, we have χ (G) + 1. Since any acyclic edge colouring is also proper, we have a (G) χ (G). It has been conjectured by Alon, Sudakov and Zaks [2] that a (G) + 2 for any G. In fact, unless G is a complete graph of even order or a graph which is almost complete (in terms of the number of edges it contains) it is believed that a (G) (G) + 1. Using probabilistic arguments, Alon, McDiarmid and Reed [1] proved that a (G) 64. The best known result, as of now, for arbitrary graphs, is by Molloy and Reed [13] who showed that a (G) 16. Muthu, Narayanan and Subramanian [14] proved that a (G) 4.52 for graphs G of girth at least 220 (girth is the length of a shortest cycle in a graph). Though the best known upper bound for the general case is far from the conjectured value of + 2, the conjecture has been shown to be true for some special classes of graphs. Alon, Sudakov and Zaks [2] proved that, there exists a constant k such that a (G) + 2 for any graph G whose girth is at least k log. They also proved that a (G) + 2 for almost all -regular graphs. This result was improved by Nešetřil and Wormald [17] who showed that for a random -regular graph a (G) + 1. From Burnstein s [7] result we know that the conjecture is true for subcubic graphs. Skulrattankulchai [18] gave a polynomial time algorithm to colour a subcubic graph using colours. Muthu, Narayanan and Subramanian proved the conjecture for grid-like graphs [15] and outer planar graphs [16]. In fact they gave a better bound of + 1 for those classes of graphs. Here, we verify the conjecture for the class of 2-degenerate graphs, and the stronger conjecture for some subclasses of this class. We state our results in section 2, and the proofs appear in the subsequent sections.

2 2 Our Results Here, we formally define the classes of graphs we handle and state the results we obtain. Definition 1. A 2-degenerate graph is any graph, each of whose subgraphs has a vertex of degree at most 2. Definition 2. A series-parallel graph is any simple graph obtained starting from K 2 and performing any sequence of the following two operations: (i) subdivide an edge (ii) add edges parallel to existing edges. Finally all multiple edges are eliminated to render the graph simple. Definition 3. A 2-tree is a graph recursively defined as follows. A K 3 (triangle) is a 2-tree. Given a 2-tree T n on n vertices, to construct another 2-tree on n+1 vertices, add a new vertex and make it adjacent to the endpoints of an edge (K 2 ) of T n. With respect to the above definition of 2-trees, we define some terms and notation which we will use later. The triangle with which the construction is initiated is called the initial triangle or base triangle. The set of all ears added through the endpoints of a given edge (u, v) is denoted ext(u, v). (u, v) is called the base edge for each of these ears. Definition 4. A partial 2-tree is any subgraph of a 2-tree. From the definition it is easily seen that 2-trees are triangulated, planar and also 2-degenerate. We denote the class of all 2-trees by T and the class of partial 2-trees by P. We prove the conjecture for 2-degenerate graphs. A graph G is called k-degenerate if every subgraph of G, has a vertex of degree at most k. For example, planar graphs are 5-degenerate, forests are 1-degenerate. 2-degenerate graphs properly contain series-parallel graphs, outerplanar graphs and planar graphs of girth at least 6 as subclasses. The earliest result on acyclic edge colouring of 2-degenerate graphs was by Card and Roditty [8], where they proved that a (G) + k 1, where k is the maximum edge connectivity, defined as k = max u,v V (G) λ(u, v), where λ(u, v) is the edge- connectivity of the pair u,v. Note that here k can be as high as. Muthu, Narayanan and Subramanian [16] proved that a (G) + 1 for outerplanar graphs which is a subclass of 2-degenerate graphs and posed the problem of proving the conjecture for 2-degenerate graphs as an open problem. In this paper, we prove the following theorems, Theorem 1. If G is a 2-degenerate graph, then a (G) + 2. Theorem 2. If G is a series-parallel graph, then a (G) + 1. Theorem 3. If G is a 2-tree or a partial 2-tree, then a (G) + 1. Actually, it is known that the class of series-parallel graphs is strictly contained in the class of partial 2-trees (see [10]), and so Theorem 2 follows as a consequence of Theorem 3. We however give independent proofs of both since the proof methods differ radically. A lot of the work in this field has been nonconstructive, using probabilistic methods. In contrast, our proof is constructive and yields an efficient polynomial time algorithm. Determining a (G) is a hard problem both from a theoretical and from an algorithmic point of view. Even for the simple and highly structured class of complete graphs, the value of a (G) is still not determined exactly. It has also been shown by Alon and Zaks [3] that determining whether a (G) 3 is NP-complete for an arbitrary graph G. It follows from the reduction used in [3] that it is in fact NP-hard to determine a (G) even when G is a 2-degenerate subcubic graph. The vertex version of this problem has also been extensively studied (see [5], [7], [6]). A generalisation of the acyclic edge chromatic number has also been studied. See [9], [11] for further reading. 3 Proof of Theorem 1 We now specify some notation which we will use in the following 2 sections. Let G = (V, E) be a graph. Let x V. Then N G (x) will denote the set of neighbours of x in G. For S V, G[S] will denote the subgraph of G induced by S. For an edge e E, G e will denote the graph obtained by deletion of the edge e. For x, y V, when e = (x, y) = xy, we may use G {xy} instead of G e. Let C : E {1, 2,..., t} be an acyclic edge colouring of G. For an edge e E, C(e)

3 S V (G) \ (S {x}) N (x) N (x) x y Figure 1: 2-degenerate graph will denote the colour given to e with respect to the colouring C. For x, y V, when e = (x, y) = xy we may use C(x, y) instead of C(e). We use the notation d G (v) to denote the degree of the vertex v in graph G. If G is clear from the context we abbreviate this to d(v). δ(g) or δ will be used to denote the minimum degree of the graph. Proof. We prove the theorem by induction on the number of edges. Let G be a 2-degenerate graph with n vertices and m edges. Assume the theorem is true for all graphs with at most m 1 edges. To prove the theorem for G, we may assume that G is connected. We may also assume that δ(g) 2, since otherwise if there is a vertex v, with d(v) = 1, we can easily extend the acyclic edge colouring of G e (where e is the edge incident on v) to G. Let S = {v V (G) : d(v) = 2}. Since G is 2-denegerate S. Since G[V S] is also 2-degenerate, there exists at least one vertex of degree at most 2 in G[V S]. Let x be one such vertex. Let N G (x) = N G(x) S and N G (x) = N G(x) N G (x). Note that N G (x) represents exactly the neighbours of x in G with degree strictly greater than 2. It is easy to see that N G (x) 2 and N G (x) 1. Let y N G (x) (see Figure 1 ). Let G = G {xy}. It follows that N G (x) = N G (x) {y}, N G (x) = N G (x) {y} and N G (x) = N G (x). By induction on the number of edges, graph G is + 2 acyclically edge colourable. Let C : E {1, 2,..., + 2} be an acyclic edge colouring of G. Let F x = {C(x, z) z N G (x)}. Let F x = {C(x, z) z N G (x)} and F x = {C(x, z) z N G (x)}. Note that F x is the disjoint union of F x and F x and also F x 2. Let a be the only neighbour of y, in G. Let S a = {C(a, z) z N G (a ) {y}}. A colour α is a candidate for an edge e if none of the incident edges of e are coloured α. A candidate colour α is valid for an edge e if assigning the colour α to e does not result in any bichromatic cycle. Our aim now is to extend the acyclic edge colouring C of G to G by giving a colour to the edge xy from the available + 2 colours. If (G ) < (G), then the colouring C only uses (G ) + 2 (G) + 1 colours. A new colour could be used to colour edge xy in G, thereby completing the induction. Hence we assume (G ) = (G) =. Since F x {C(y, a )}, we have at least 2 candidate colours for the edge xy. case 1: C(y, a ) / F x Then clearly all the candidate colours are valid for the edge xy, since any cycle involving the edge xy will contain the edge ya as well as an edge incident on x in G and thus the cycle has at least 3 colours. case 2: C(y, a ) F x Here we have two subcases. case 2.1: C(y, a ) F x Let a N G (x) be the vertex such that C(y, a ) = C(x, a). Let z be the only neighbour of a other than x. It is easy to see that, when we colour edge xy there is a possibility of a bichromatic cycle only if we assign C(x, y) = C(a, z). But since we have at least 3 candidate colours for edge xy, we can easily avoid this situation. case 2.2: C(y, a ) F x Here we show that the colouring C of G can be modified such that this case reduces to one of the previous cases. Since S a {C(y, a )}, we have at least 2 candidate colours which can be used to replace the colour on the edge ya. Note that any candidate colour is valid for the edge ya in G since y is a pendant vertex in G. Recall that F x 2 and since C(y, a ) F x, at most one of these candidate colours can be in F x. Thus there is at least one colour which can be used to replace the colour on the edge ya such that this case reduces to either case 1 ( i.e., C(y, a ) / F x ) or case 2.1 ( i.e., C(y, a ) F x ).

4 4 Series-Parallel Graphs Here, we give the proof of Theorem 2. The proof ideas are similar to those used in the previous section. We also use notation similar to that used there. Proof. We prove the theorem by induction on the number of edges. Let G be a series-parallel graph with n vertices and m edges. Assume the theorem is true for all series-parallel graphs with at most m 1 edges. To prove the theorem for G, we may assume that G is connected. We may also assume that δ(g) 2, since otherwise there is a vertex v, with d(v) = 1 and we can easily extend the acyclic edge colouring of G e (where e is the edge incident on v) to G. The set S and the vertex x are defined as used in the previous section. Also N G (x) and N G (x) are defined as before. It follows that N G (x) 2 and N G (x) 1. The following 2 claims will help us focus on essentially the hardest parts of the proof of Theorem 2 as they dispense with the simpler cases. Claim 1. For any a, b S, if (a, b) E, the colouring can be extended easily by induction. Proof. Let a and b be the only neighbours of a and b respectively in G {ab}. Now we can extend the acyclic edge colouring C of G {ab} to G. This is easy to see because if C(a, a ) C(b, b ), then there are 1 (note that we can assume that this value is at least 1) candidate colours, which are also valid for edge ab, otherwise if C(a, a ) = C(b, b ), then there are candidate colours for the edge ab and among them at least one is valid as N G (a ) a 1. Claim 2. For any a S and a, b N G (a), if (a, b ) / E, the colouring can be extended easily by induction. Proof. Let G = G a + a b. Note that G is also series-parallel. Now G has m 1 edges, and by induction hypothesis it has an acyclic edge colouring C using at most + 1 colours. We can extend this colouring to G by assigning colour C(a, b ) to edge aa and a missing colour at b with respect to colouring C to the edge ab. Note that this colouring does not create any bichromatic cycles as the colour of edge aa (i.e., C(a, b ) in C) is not present at b in G. In view of Claims 1 and 2, we will assume S is an independent set and that the neighbourhood of any vertex in S induces a clique. Let y N G (x) (see Figure 1 for reference). Let G = G {xy}. It follows that N G (x) = N G (x) {y}, N G (x) = N G (x) {y} and N G (x) = N G (x). By induction on the number of edges, graph G is + 1 acyclically edge colourable. Let C : E {1, 2,..., + 1} be an acyclic edge colouring of G. Let F x = {C(x, z) z N G (x)}. Let F x = {C(x, z) z N G (x)} and F x = {C(x, z) z N G (x)}. Note that F x is the disjoint union of F x and F x and also F x 2. Let a be the only neighbour of y, in G. Let S a = {C(a, z) z N G (a ) {y}}. Our aim now is to extend the acyclic edge colouring C of G to G by giving a colour to the edge xy from the available + 1 colours. If (G ) < (G), then the colouring C only uses (G ) + 1 (G) colours. A new colour can be used to colour edge xy in G, thereby completing the induction. Hence we assume (G ) = (G) =. Since F x {C(y, a )}, we have at least one candidate colour for the edge xy. case 1: C(y, a ) / F x Then clearly all the candidate colours are valid for the edge xy, since any cycle involving the edge xy will contain the edge ya as well as an edge incident on x in G and thus the cycle has at least 3 colours. case 2: C(y, a ) F x Here we have two subcases. case 2.1: C(y, a ) F x Let a N G (x) be the vertex such that C(y, a ) = C(x, a). Let z be the only neighbour of a other than x. It is easy to see that, when we colour edge xy there is a possibility of a bichromatic cycle only if we assign C(x, y) = C(a, z). But since we have at least 2 candidate colours for edge xy, we can easily avoid this situation. case 2.2: C(y, a ) F x Here we show that the colouring C of G can be modified such that this case reduces to one of the previous cases. Since S a {C(y, a )}, we have at least one candidate colour which can be used to replace the colour on the edge ya. Note that any candidate colour is valid for the edge ya in G since y is a pendant vertex in G. Recall that F x 2 and by Claims 1 and 2, and the fact that C(y, a ) F x \ S a in this case, it follows that F x S a = 1. Thus, none of these candidate colours for ya can be in F x. Thus there is at least one colour which can be used to replace the colour on the edge ya such that it reduces to either case 1 ( i.e., C(y, a ) / F x ) or case 2.1 ( i.e., C(y, a ) F x ).

5 5 2-trees In this section we give the proof of Theorem 3 for 2-trees. An extension of this idea yields the result for partial 2-trees. This will appear in Section 6. We use L v to denote the subset of colours not seen by the vertex v in any partial colouring of G. Note that L v U v +1 (since we are using + 1 colours), where U v is the number of uncoloured edges incident to v. d(v) denotes the degree of the vertex v. The following lemma describe an acyclic edge colouring for the complete bipartite graph, K 2,t, using t colours if t 3, and using t + 1 colours otherwise. Lemma 1. For the complete bipartite graph, K 2,t, { t if t 3 a (K 2,t ) = t + 1 otherwise. Proof. Let u and v be the vertices in the partite set of size 2, and let w 1,...,w t be the vertices of the partite set of size t. If t = 1 or 2, the colouring is straightforward. If t 3, colour (u, w i ) with colour i, for i {1,...,t} and colour (v, w i ) with colour i +1, for i {1,..., t 1} and colour (v, w t ) with colour 1. It is easy to observe that the subgraph induced by any pair of colour classes is a path on 3 or 4 edges, or a collection of 2 vertex disjoint paths on 2 edges each. In each case, the subgraph is acyclic and hence the colouring is proper and acyclic. We call this scheme colouring by shifting and will use it as a subroutine in the colouring of partial 2-trees. Also, note that no maximal 2-coloured path ends at u or v in this scheme. We now describe a generalised version of the previous lemma. Lemma 2. Consider the complete bipartite graph H = (A, B, F) with A = {u, v} and B = {w 1,..., w t } where t 3. Let L u denote a set of t colours which are permitted for edges incident at u. L v is defined similarly. Then, there is an acyclic edge colouring of H using only colours from L u and L v for edges incident at u and v respectively. Proof. Without loss of generality, assume that I = L u L v = {1,..., i} is the set of i 0 colours available for edges incident at both u and v and also that L u \ I = {i + 1,...,t} and also that L v \ I = {t + 1,...2t i}. Then, colour the edges (u, w 1 ),...(u, w t ) with 1,...,t respectively. Colour the edges (v, w 1 ),..., (v, w i 1 ) with 2,...,i respectively. Colour the edges (v, w i )...,(v, w t 1 ) with t + 1,...,2t i respectively and (v, w t ) with colour 1. It is easy to observe that the subgraph induced by any pair of colour classes is acyclic and hence the colouring is proper and acyclic. We call this scheme also colouring by shifting and will use it as a subroutine in the colouring of partial 2-trees. We may assume that the given graph is biconnected, since an acyclic colouring of any graph can be obtained from acyclic colouring of its biconnected components using a (G) colours. First we prove the result for the class T of 2-trees and then extend it, in the next section, to include all of P, the partial 2-trees. We will use the following easy to verify fact repeatedly in our proofs. Observation 1. If G is a 2-tree, one can construct G from any arbitrary triangle of G by adding 2-ears according to the definition of 2-trees stated in Section 2. We obtain a + 1 acyclic edge colouring of any 2-tree by an iterative colouring procedure which incorporates more edges at each stage into an existing partial colouring until the graph is fully coloured. There is in general more than one way in which a 2-tree can be constructed according to the definition given. We will describe a particular order in which the vertices and edges are introduced and use this ordering to colour the graph. The procedure we give never alters the colour of an edge once it has been assigned. 5.1 The colouring procedure We reconstruct the graph G by starting from any triangle T = {a, b, c} as mentioned in Observation 1 by building the graph ear by ear. We call this the base triangle or initial triangle. Recall that ext(u, v) denotes the set of all 2-ears having (u, v) as its base edge. We also assign a nonnegative integer value to each edge e and call it the level of e and denote it by level(e).

6 G k 1 ears u x a y w v k 2 ears Figure 2: 2-tree Initially, the three edges of T are assigned level 0. The level number is defined inductively. If e = (u, v) is any edge already added such that level(e) = i, then for each 2-ear (u, w, v) ext(u, v), we assign level(u, w) = level(w, v) = i + 1 and add this 2-ear. In addition, we we follow the convention that (i) edges are added in increasing order of their level numbers and (ii) if (u, w, v) is a level i ear, then all (i+1)-level ears of ext(u, w) appear contiguously (the same holds for ext(w, v) also) with 2-ears of one set appearing immediately before or after the 2-ears of the other set. When we are going to add ears of level i + 1, the graph will look similar to Figure 2. This is the order in which the edges would be introduced and also coloured. In the following, we use to denote the maximum degree of the current graph (after adding the edges to be coloured at this step). The colouring procedure can be summarised as follows. 1. Colour the base triangle T = {a, b, c} with colours 1, 2, Colour the level 1 edges in the three sets ext(a, b), ext(b, c) and ext(a, c), using the shifting procedure for each of the sets of ears. Notice that a set of k ears introduced with the same base edge constitute a complete bipartite graph K 2,k. 3. For i 1, the procedure for colouring level-(i + 1) edges is as follows. Assume that all edges up to level-i have already been added and coloured (using + 1 colours). Assume that for some level i 2-ear (u, w, v), we are adding the edges in ext(u, w) and ext(w, v). We refer to (u, w, v) as the base ear. Please refer to Figure 2. Let the number of 2-ears in ext(u, w) and ext(w, v) be, respectively, k 1 and k 2. We assume without loss of generality that k 1 k 2. Colour the new edges as described below, under colour extension. Colour Extension We describe below how to extend the colouring C to the newly added ears. The procedure falls under a number of cases. We colour the ears in ext(u, w) first and then those in ext(v, w). Let C(uw) = x and C(vw) = y and C(uv) = a. Notice that since we use + 1 colours, we have L u k 1 + 1, L v k and L w k 1 + k One should also note that L u L w k 1. Case k 1 = 0: If K 2 = 1, colour the edge of the ear incident to w using C(uv) = a and other edge with any colour from L v \ {x}. Similarly if K 2 = 2, colour one ear with 2 colours from L v \ {x} and the other ear with one of these colours and a. Otherwise the K 2 ears based on edge (v, w) are coloured using k 2 colours from L v \{x} using the shifting procedure.

7 Case k 1 = 1: In this case, we color the single ear based on (u, w) with the colours a and a colour from L u \ {y}. For the ears based on (v, w), we pick a subset of k 2 colours from L v \ {x} of cardinality k 2 and a set of same size from L w and colour using the shifting procedure. Case k 1 = 2: In this case, one of the ears based on (u, w) is coloured using 2 colours from L u \ {y} while the other is coloured using a and one of these colours. For the ears based on (v, w), we pick a subset of k 2 colours from L v \ {x} of cardinality k 2 and a set of same size from L w and colour based on shifting. Case k 1 3: In this case, the ears based on (u, w) are coloured using k 1 colours from L u \ {y} in shift. Observe that here, L w L v, so the selected set of colours are free at both endpoints of the ear set. For the ears based on (v, w), we pick a subset of k 2 colours from L v \ {x} of cardinality k 2 and a set of same size from L w and colour based on shifting. Observe that from the description of the first colouring by shifting procedure, after the first sets of k 1 ears are coloured in the last case above, there is no maximal 2 coloured path between u and w. Thus any bichromatic cycle created as a result of adding the second set of ears must necessarily pass through the edge (u, w). This is not possible because we do not use the colour x of that edge in the colouring of the second set of ears. From Lemmas 1 and 2, we know that the colouring by shifting procedures never creates bichromatic cycles among themselves. The first 3 cases are more straightforward, and it can be seen that the colouring is proper and acyclic in each case. 6 Partial 2-tree Here, we extend the proof given above to partial 2-trees. Given any partial 2-tree T, we consider one of the minimum 2-trees in which this 2-tree is contained and call it G. We mark all the edges of G which are not in T as imaginary edges. We use the imaginary edges only to classify the level of edges for the further addition of ears. They do not contribute to the degree of a vertex in G. They are never coloured. Thus d G (v) = d T (v). The important point to notice is that, again we need only (T) + 1 colours to extend the colouring. An ear consisting of both real edges is called a full ear, while ears with one real edge and one imaginary edge are called half ears. Observe that empty ears (both edges are imaginary) are inconsequential, since we do not colour them at all, and only use their endpoints for the addition of higher level ears. Suppose, at any point, we are to colour k 1, k 2 pairs of ears (some of them are half ears). We notice that if there are k uncoloured real edges at an endpoint, then we have at least k + 1 available colours for the edges incident at the endpoint. Here the ears are ordered with all the full ears first followed by the half ears and finally by the empty ears. Colour the full ears as mentioned earlier for 2-trees and extend the colouring to partial ears in a proper fashion. It follows that such a colouring is proper and acyclic. It is identical to the case of 2-trees, except for the half ears. However, half ears only give rise to pendant edges and cannot create bichromatic cycles, so any proper colouring is sufficient. This completes the proof of Theorem 3 for partial 2-trees. References [1] N. ALON, C.J.H.MCDIARMID, AND B.A.REED, Acyclic coloring of graphs, Random Structures and Algorithms, 2 (1991), pp [2] N. ALON, B. SUDAKOV, AND A. ZAKS, Acyclic edge-colorings of graphs, Journal of Graph Theory, 37 (2001), pp [3] N. ALON AND A. ZAKS, Algorithmic aspects of acyclic edge colorings, Algorithmica, 32 (2002), pp [4] D. AMAR, A. RASPAUD, AND O. TOGNI, All to all wavelength routing in all-optical compounded networks, Discrete Mathematics, 235 (2001), pp [5] B.GRÜNBAUM, Acyclic colorings of planar graphs, Israel Juornal of Mathematics, 14 (1973), pp

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