Finding a five bicolouring of a triangle-free subgraph of the triangular lattice.

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1 Finding a five bicolouring of a triangle-free subgraph of the triangular lattice. Frédéric avet and Janez Žerovnik July 8, 1999 Abstract A basic problem in the design of mobile telephone networks is to assign sets of radio frequency bands (colours) to transmitters (vertices) to avoid interference. ften the transmitters are laid out like vertices of a triangular lattice in the plane. We investigate the corresponding colouring problem of assigning sets of colours of size to each vertex of the triangular lattice so that the sets of colours assigned to adjacent vertices are disjoint. A - colouring of a graph is a mapping from into the set of the subsets of such that and for any adjacent vertices and, "!. We give here an alternative proof of the fact that every triangular-free induced subgraph of the triangular lattice is 5-[2]colourable. This proof yields a constant time distributed algorithm that finds a 5-[2]colouring of such a graph. We then give a distributed algorithm that finds a #%$'& (*),+ colouring of a triangle-free induced subgraph of the triangular lattice with at most -.0/ colours. 1 Introduction. A basic problem in the design of mobile telephone networks is to assign sets of radio frequency bands (colours) to transmitters (vertices) to avoid interference. The number of bands demanded at transmitter 4 may vary between transmitters. We assume that the transmitters are located like vertices in a triangular lattice in the plane: this pattern is often used as it gives a good coverage. We assume also that adjacent vertices must not be assigned the same band, so as to avoid interference. There are more refined versions of this channel assignment problem, see for example [3, 4], in which we insist on a minimum separation between channels assigned to two transmitters (where this minimum separation depends on the proximity of the transmitters). But we consider only the most basic case here. The channel assignment problem described above is a weighted colouring problem on the triangular lattice. A weight function on a graph 8 is a function from 9:2;8<, the set of vertices of 8, into the set of non-negative integers. Let us denote the set >?A@7?B'BB?DCFE by G>?ICKJ and 1

2 ] let 1 be a weight function on a graph 8. A C -G 1 J colouring of 8 is a mapping from 9 2;8< into the set of the subsets of G>?IC J such that and for any adjacent vertices and 4, The graph 8 is C -G 1 J colourable if it admits an C -G 1 J colouring. The G 1 J chromatic number of a graph 8, denoted by 258, is the smallest integer C such that 8 is C -G 1 J colourable. If 2;8< C, we say that 8 is C -G 1 J chromatic. Let be a positive integer. For convenience, we denote by the weight function such that 2 47 for each vertex 4 of 8. The [1]colouring is the usual colouring: to each vertex, we associate a colour in such a way that two adjacent vertices become different colours. In this paper, we call [2]colourings bicolourings, and [3]colourings tricolourings. There is a natural graph 8 associated with a pair 2;8?%1K as above, obtained by blowing up each vertex 4 by a clique on vertices. Weighted G 1 J colourings of 8 correspond to usual colourings of the graph 8, so 2;8< 258. More generally, C -G 1 J colouring of the graph 8 is equivalent to C -G 1 J colouring of the graph 8. So I. Moreover 2;8< 2;8<! "# 258 2%$,. The G 1 J weighted clique number of 8, denoted by &', is (*),+.-0/ ?25798%:<;>?A@CB 8 E. Clearly, & 2;8< & 2;8D. Thus EF&G. We are interested in the G 1 J chromaticity of an induced subgraph of the triangular lattice, as this corresponds precisely to the basic channel assignment problem described above. This lattice graph may be described as follows. The vertices are all integer linear combinations JILKM of the two vectors I 2>?2N and M 2, P?JQ P R : thus we may identify the vertices with the pairs 2?2K of integers. Two vertices are adjacent when the Euclidean distance between them is 1. Thus each vertex S 2?TK has the six neighbours: its left neighbour 2 VU >?2K, its right neighbour 2 >?2K, its leftup neighbour 2 U >?2KW >, its rightup neighbour 2?2K >, its leftdown neighbour 2?2K U > and its rightdown neighbour 2 >?2K U >. There is an obvious 3-colouring of the infinite triangular lattice which gives rise to the partition of the vertex set of any triangular lattice graph into three independent sets, Red, Blue and Green such that if S is in Red (resp. Blue or Green) then its right neighbour is in Blue (resp. Green or Red). According to this partition each vertex is called red, blue or green. McDiarmid and Reed [5] proved that for every induced subgraph of the triangular lattice and every weight function 1 : &Y > A distributed algorithm which guarantees the &!\[ [7]. McDiarmid and Reed conjectured that this ratio [ bound is reported by Narayanan and Schende can be improved: Conjecture 1 (McDiarmid and Reed) There is a constant such that for every induced subgraph of the triangular lattice and any weight function 1 2;8< &G 2;8< ^ B 2

3 2 P ^ R is best possible: the cycle on 9 vertices, 5 is an induced subgraph of the ] Ẍ ] ^ triangular lattice and 2 5A [ & 9[. We first investigate this conjecture when the weight function is a constant. The triangular This ratio ] [ lattice is -G J chromatic. Therefore, any of its subgraphs is -G J colourable. If it contains a triangle, then it is -G J chromatic and &. So &. So we just need to study the G J chromaticity of triangle-free induced subgraphs of the triangular lattice. Let 2 be the maximum G J -chromatic number of such a graph. The above conjecture of Reed and McDiarmid restricted to [k]colouring is the following: Conjecture 2 (McDiarmid, Reed) There is a constant such that 2 -. avet [2] proved that every triangle-free induced subgraph of the triangular lattice is 7- tricolourable. This implies that 2. But his proof did not yield any distributed algorithm to find such a G J colouring. In the next section, we expose an alternative proof of Theorem 1 of [2] that states that every triangle-free induced subgraph of the triangular lattice is 5-bicolourable. # This proof gives a distributed algorithm which finds a 5-bicolouring (and then a U G J colouring) of such a graph. #%$& In the third section, using this algorithm, we exhibit a distributed algorithm which finds a - -[p]colouring of a triangle-free induced subgraphs of the triangular lattice. 2 5-bicolouring of the triangle-free induced subgraphs of the triangular lattice. Lemma 1 (avet, [2]) Let 2 S? S? S P?'BBB'? S be a path of length and and two 2-subsets of G>?J. There exists a 5-bicolouring of such that 2 S and 2 S if and only if: > and K2? P >, and K2? P E >, or E For the sake of completeness we give a sketch of proof here. Note that the graph with vertices being 2-subsets of the set G>? J in which two sets are adjacent if and only if they have empty intersection is isomorphic to the famous Petersen graph. As it is well-known that in the Petersen graph any pair of vertices lies on a 5-cycle, the assertion of the above lemma follows. 8 8 Theorem 1 Let be a triangle-free induced subgraph of the triangular lattice. Then is 5-bicolorable. Proof. A vertex of 8 is said to be suitable if its left, rightup and rightdown neighbours in the lattice do not belong to 8. It is easy to see that two suitable vertices are not adjacent in 8. Let be the set of suitable vertices of 8. Let be the bicolouring of the vertices of defined as follows: if 4 is a red vertex, 2 47 >?A@ E, 3

4 @? R if 4 is a blue vertex, E, if 4 is a green vertex, 2547 >? E. We will prove that this bicolouring of may be extended to a 5-bicolouring of 8. A path 2 S? S P?BB'B'? S is left (resp. rightup, rightdown) if S is the left (resp. rightup, rightdown) neighbour of S in the triangular lattice. A tristar is the union of one left, one rightup and one rightdown path emerging from a common origin. Let 8 be the graph induced by 8 on the vertices of 9:2;8<. It is easy to see that the components of 8 are tristars and only the terminus of the left (resp. rightup, rightdown) path may have a left (resp. rightup, rightdown) neighbour in. Let be a tristar of 8. Let 2 S3?? P?BBB?T be the left path of, 2 S3?I4?I4 P?BB'B'?I4 the rightup path of and the 2 S3?? P?BB'B'? rightdown path of. And let be the left neighbour of, 4 the rightup neighbour of 4 and the rightdown neighbour of. Let 8 be the subgraph induced by 8 on 9:2, 3?I4? <E. We shall prove that there exists a 5-bicolouring of 8 that coincides with on, 3?I4? E. Suppose first that two paths of the tristar suppose that If then in 8, the path have the same length. By symmetry, we may 2.?BBB? S3?I4?BBB?I4 5?D4 has length at least. Then 4 and have the same label (red, blue or green) and so ?T?T four. So, by Lemma 1, there exists a 5-bicolouring of such that 2 2 and Thus setting 2 254, the 5-bicolouring of 8 coincides with on,?d4k? <E. If >, then in 8, 2 3?BBB? S3?BB'B'?I47 is a path of length three. Furthermore, and 4 have different labels, and hence Thus, by Lemma 1, there exists a 5-bicolouring of such that 2 2 and Setting 2 254, the 5-bicolouring of 8 coincides with on, 3?I4? E. If N, then N. So is a single vertex S with three (possible) neighbours in 8,, 4 and. Now,, 4 and have the same label so Setting 2 S be a 2-subset of G >?J disjoint from 2, we have the result. Suppose now that the three paths of have distinct lengths. By symmetry, we may suppose that!". We will prove the result by induction on. If then > and N. If is red, then 4 is blue and is green. The colouring defined as follows is suitable: 2 P E, >? E, 2 E. If is blue, then 4 is green and is red. The colouring defined as follows is suitable: 2 P >? E, 2 E, 2 S? E. If is green, then 4 is red and is blue. The colouring defined as follows is suitable: 2 E, 2 254? E, 2 S?>E. If FE, by induction hypothesis, there exists a 5-bicolouring of U,?T? # P E such that and ?T P? #?T?T is a path of length four. So, by Lemma 1, one can extend in a 5-bicolouring of that coincides with on, 3?I4? <E. $ The proof of the above theorem gives the following distributed algorithm for finding a bicolouring of an induced subgraph of the triangular lattice which runs in constant time. We may suppose that each vertex knows its label. And that each fourth vertex on a line is special. These two assumptions make sense because they are fixed by the triangular lattice and do not depend 4

5 - on the triangle-free graph we want to 5-bicolour. Step 1: Colour every red (resp. blue, green) suitable vertex by E E, >? E ). This can be done in one communication unit of time, because a vertex only need to know its neighbours to decide whether it is suitable or not. Step 2: Bicolour arbitrarily every special vertex that is at distance at least four of all suitable vertices and all centers of tristars. This can be done in four communication units of time. Every suitable vertex and every center of tristar send a message that is transmitted from neighbour to neighbour. Each special vertex that has received no message, four units of time after is given an arbitrary 2-set of colours. Step 3: Extend the bicolouring to the whole graph. This is possible according to Lemma 1 and the proof of Theorem 1. Indeed, this can be done in at most 1 units of communication time : the graph induced by the vertices which are not bicoloured after Step 2 is the union of paths of length at most two and tristars which are unions of paths of length at most seven. So it requires at most 1 units of time for a non-coloured vertex to completely know its component and the bicouloring of the neighbours of the endvertices of its component. 3 [p]colouring of the triangle-free induced subgraphs of the triangular lattice. - > > ? 2? U N 2? 2?? 4 N > We now give a distributed algorithm that finds a #%$'&(),+ [p]colouring of a triangle-free induced subgraph of the triangular lattice. We suppose that each vertex of the graph is already labelled by Red, Blue or Green according to the 3-colouring of the triangular lattice. Step 0: Initialize, and. Step 1: Assign to each red (resp. blue, green) vertex of such that 2547, the colour Red, (resp. Blue, Green). Step 2: Let be the graph induced by on the set of vertices for which Assign to each isolated vertex of the colours K?>,?A@,, and? (or K?>?'BBB'2?T 2547 if Let 47 (or 2547 if 2547 ). Step 3: Let be the graph induced by the vertices of which are not isolated. Colour each of these vertex by two colours in 72?>,?A@,,,2 AE by the previous algorithm. For a vertex of, let Step 4: Let be the graph induced by the vertices of such that If is not empty, do,,, and go to Step 1. It is easy to see that this algorithm yields a #%$'&'(*),+ [p]colouring of 8. Indeed it is easy to see that for each loop from Step 1 to Step 4, & %2 & '20 U $'& (*),+. Thus the algorithm makes at most - loops. And the algorithm uses 3 special colours Red, Blue and Green and 5 colours #%$ & (*),+ at each loop. So the algorithm uses at most - colours. Note that this algorithm uses that many colours only if two vertices of an edge have weight &# 258 [@. therwise this algorithm 5

6 - will use fewer colours. In particular, if the vertices of a stable set of 8 have huge weight and the other vertices have small weight then this algorithm will use roughly 8< colours. Each loop of the above algorithm can be done simultaneously. This yields the following algorithm: Step 1: Assign to each red (resp. blue, green) vertex of 8 with Red (resp. Blue, Green). Let * [@. Step 2: Find a 5-bicolouring of 8. Step 3: Let 4 be a vertex of 8,?2KE 2547 and 2 4 Set 2547 [@. the colour ( )\+, 2 A? neighbour of 4,E [@ and If , assign to 4 the colours 2D? and 2D?2K for >. If assign to 4 the colours 2D? and 2?2K for > A0. Let U. Assign to 4 the colours 2D?> 2D?A@, 2?, 2D? and 2D? for and the colours 2 >?> BBB'2 >?F. Since the algorithm for finding a 5-bicoloration is distributed and runs in constant time, so does the above algorithm. ence, Theorem 2 Let 8 be a triangle-free subgraph of triangular lattice and 1 an arbitrary weigh function. Then there is a constant time distributed algorithm which finds a G 1 J colouring of 8 using at most #%$'&(),+ colours. References [1] J.R. Griggs and D. Der-Fen Liu, The channel assignment problem for mutually adjacent sites, J. Combinatorial Theory A 8 (1994) [2] F. avet, Channel assignment and multicolouring of the induced subgraphs of the triangular lattice, manuscript. [3] W.K. ale, Frequency assignment, Proceedings of the IEEE, 8 (1980) [4] J. van den euvel, R.A. Leese and M.A. Shepherd, Graph Labelling and Radio Channel Assignment, J. Graph Theory 29 (1998) [5] C. McDiarmid and B. Reed, Channel assignment and weighted colouring, manuscript. [] C. McDiarmid and B. Reed, Colouring proximity graphs in the plane, Discrete Math. 199 (1999)

7 [7] L. Narayanan and S. Schende, Static Frequency Assignment in Cellular Networks, Sirocco 97, (Proceedings of the 4th international Colloqium on structural information and communication complexity, Ascona, Switzerland), D. Krizanc and P. Wildmayer (eds.), Carleton Scientific 1997, pp [8] L. Narayanan, N. Schabanel, S. Ub éda and J.Žerovnik, A note on upper bounds for the span of the frequency planning in cellular networks, manuscript. 7

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