Labelling Wheels for Minimum Sum Number. William F. SMYTH. Abstract. A simple undirected graph G is called a sum graph if there exists a

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1 Labelling Wheels for Minimum Sum Number Mirka MILLER Department of Computer Science University of Newcastle, NSW 308, Australia SLAMIN Department of Computer Science University of Newcastle, NSW 308, Australia Joseph RYAN Department of Mathematics University of Newcastle, NSW 308, Australia William F. SMYTH Department of Computer Science and Systems McMaster University, Hamilton, Ontario, Canada School of Computing, Curtin University Bentley, WA 610, Australia Abstract A simple undirected graph G is called a sum graph if there exists a labelling of the vertices of G into distinct positive integers such that any two distinct vertices u and v of G are adjacent if and only if there is a vertex w whose label w) = u) + v). It is obvious that every sum graph has at least one isolated vertex, namely the vertex with the largest label. The sum number H) of a connected graph H is the least number r of isolated vertices K r such that G = H + K r is a sum graph. It is clear that if H is of size m, then H) m. Recently Hartseld and Smyth showed that for wheels W n of order n+1 and size m = n, W n ) m); that is, that the sum number is of the same order of magnitude as the size of the graph. In this paper we rene these results to show that for even n 4, W n ) = n= +, while for odd n 5 we disprove a conjecture of Hartseld and Smyth by showing that W n ) = n. Labellings are given that achieve these minima. 1 Introduction Since the introduction of sum graphs by Harary [4], there have been several papers specifying or bounding the sum number of particular classes H of graphs: complete graphs [1], complete bipartite graphs [5], complete multipartite graphs [7], and trees []. In all of these cases, H) m=n), where n is the order number of vertices) and m the size number of edges) of H; that is, the order of magnitude of the sum number is at most the order of the graph. It is known however [3] that there exist classes H of graphs such that H) n ), even though no such graphs have yet been constructed. A step in the direction of 1

2 constructing such graphs was taken by Hartseld and Smyth [6], who showed that for wheels W n, W n ) m). Wheels are thus graphs of some interest in this context, and in this paper we extend the results of [6] to give labellings for all wheels that achieve the minimum sum number. In Section of this paper we show that for even n 4, W n ) = n +, thus correcting an error in [6]. In Section 3 we show that for odd n 5, W n ) = n, disproving the conjecture in [6] that W n ) = n +. This latter result depends on a recent paper [8] that deals with a closely-related problem, the \integral sum number" of cycles. Even Wheels For every integer n 3, a wheel W n is the graph dened by a pair of sets V; E), where V = fc; v 0 ; v 1 ; :::; v n?1 g and E = fc; v i ); v i ; v i+1 )ji = 0; 1; :::; n? 1g. The vertex c is called the centre of the wheel, each edge c; v i ), for i = 0; 1; :::; n? 1, is called a spoke, and the cycle C n = W n? c is called the rim. To simplify presentation, arithmetic on the indices of the vertices is interpreted modulo n, and we suppose that the vertices of V are already identied by their labels. In a sum graph G, a vertex w is said to label an edge u; v) if and only if w = u + v. The multiplicity of w, denoted by w), is dened to be the number of edges which are labelled by w. If w) > 0, then w is called a working vertex. If G = H + K r and H contains no working vertices, then G is said to be exclusive; otherwise, G is said to be inclusive. One of the interesting results of Hartseld and Smyth [6] is that for n odd every sum graph G = W n + K r is exclusive, while for n even every sum graph G = W n + K r is inclusive. The following lemmas will be useful for determining the sum number of even wheels. Proofs may be found in [6]. Lemma 1 Suppose that G = W n + K r is a sum graph. If for some integer i satisfying 0 i n? 1, c + v i V, then a) n is even; b) the vertices of the rim consist of n working vertices c + u j, 1 j n, which label spokes, alternating with n vertices u k, 1 k n, which do not label spokes. Lemma Suppose that G = W n + K r rim is labelled by a vertex of V. is a sum graph. Then no edge of the

3 Lemma 3 There are at least three distinct edge labels on the rim of W n. The rst lemma allows us to label the vertices on the rim of W n, for n even, alternating n working vertices with n nonworking vertices. The second and third lemmas imply that there are at least n + 3 isolated vertices required for G = W n +K r to be a sum graph unless there exists an isolated vertex that labels both a spoke and an edge of the rim. The following theorem shows that no more than two such isolated vertices can exist in G if a minimum sum number is to be achieved. Theorem 1 For n even, W n ) n +. Proof Consider a vertex t which labels both a spoke u; c) and an edge v 1 ; v ) of the rim. Then t = v 1 + v = u + c and by Lemma, t is isolated. Since G is inclusive, it follows from Lemma 1 that the vertices on the rim are alternately working and nonworking. So without loss of generality we may consider v to be a working vertex; i.e., v = v 0 + c where v 0 is a nonworking vertex on the rim. We note also that u is a working vertex else u + c would be on the rim) and may thus be expressed u 0 + c. It follows from the expression for t that u = u 0 +c = v 1 +v 0. This seems to indicate that v 1 is adjacent to both v and v 0, in contradiction to the condition of alternating working and nonworking vertices. The only labelling that manages to avoid such a contradiction is v 1 = v 0, so that v = v 1 + c and v 1 ; v 1 + c) is an edge of the rim. Now let us redene v to be the other nonworking vertex v 6= v 1 adjacent to v 1 + c. Then from the denition of a sum graph, v + c is adjacent to v 1. Similarly, the nonworking vertex v 3 6= v 1 adjacent to v + c implies the existence of a working vertex v 3 + c adjacent to v. We see that if 4 divides n, the rim vertices break down into two paths v 1 ; v + c; v 3 ; : : : ; v n= + c) and v 1 +c; v ; v 3 +c; : : : ; v n= ), while otherwise the paths are v 1 ; v +c; v 3 ; : : : ; v n= ) and v 1 + c; v ; v 3 + c; : : : ; v n= + c). Such labellings are called contrary. We see that a contrary labelling includes exactly two nonworking vertices v 1 and v n= that are adjacent to their corresponding working vertices, and that these adjacent pairs are antipodal. So far we have shown that it may be possible to reduce the sum number of G by one to n + ) by introducing the vertex t. It is clear that there can exist no more than two isolated vertices that label both spokes and edges of the rim. We show now that introducing a second such isolated vertex cannot reduce the sum number any further. Suppose that there exists a second vertex s that labels both a spoke and an edge of the rim. Then since t = v 1 + c, it follows that s = v n= + c, and so 3

4 there exist two rim vertices v 1 and v n=. It follows that n > 4. Now let x 6= v 1 and y 6= v n= denote the two other nonworking vertices adjacent to v 1 + c and v n= +c respectively. In order to label the edges of the rim, there must therefore exist vertices that are labelled with at least the following four sums: v 1 + c; v 1 + x + c; v n= + c; v n= + y + c: Further, in order to reduce the number of distinct sums to three, and so to achieve sum number n + 1, at least one pair of these sums must be identical. The only possibilities are v 1 + c = v n= + y + c and v n= + c = v 1 + x + c. But the rst of these implies that v 1 = v n= + y and so, since v 1 is a vertex, we conclude that v n= ; y) is an edge, an impossibility. Similarly the second case is impossible, and it follows therefore that there must exist at least four isolated vertices corresponding to edges of the rim. Thus the sum number of G cannot be reduced below n + by introducing s. We use the ideas in this result to construct a labelling for W n, n even, that achieves W n ) = n +. Letting n = l, we consider the vertices around the rim as two disjoint paths of length l? 1, P 1 and P, one clockwise in direction and the other counterclockwise. The centre is labelled by c, while the two paths are labelled as shown below with the parameters c, x and d, all arbitrary at this stage. Later we will formulate x in terms of d and c and give conditions under which these labels will provide a sum numbering. Let P 1 = x; x; x? d + c; x + d? c; x? d + c; x + d? c; :::; y); P = x + c; x? c; x? d + c; x + d? c; x? d + 3c; x + d? 3c; :::; z); where for l even while for l odd Alternatively, P 1 = P = 3 +?1) m l? y = x + l? z = x + l? 1 y = x? l? 1 z = x? x +?1) m d? d? d + d + l? l c; l? 1 l + 1 c; c; c:! m? 1 d??1) m d m ) l e? 1)c ; m=1 3 +?1)m )x +?1) m d m e? 1)d??1)m d m ) l e)c : m=1 4

5 In each path, each pair of adjacent vertices sum to, alternately, 3x and 3x?d+c constributing isolates to the sum number. Note that when l is even, 3x appears l l? times in each path, while 3x? d + c appears times, indicating that the nal sum is 3x. When l is odd, both sums appear l?1 times and the nal sum is 3x? d + c. We now construct the rim by connecting the two paths in a way that does not require the inclusion of any extra isolates. Joining the two initial vertices in each path requires a vertex x + c taking advantage of an already existing isolate that corresponding to the the spoke x; c). When joining the nal vertices, we note that the induced sum must be dierent to the nal sums in each path. So for l even, this nal sum must be 3x? d + c, while for l odd it is 3x. Therefore for l even we have or that is, x + l? while for l odd we have or again yielding, x? l? 1?? )d? l )c) + x + l )d? l )c) = 3x? d + c 4x + l? )d? l? 1)c = 3x? d + c )d + l? 1 x = lc? l? 1)d; )c) + x? l? 1 x? l? 1)d + lc = 3x x = lc? l? 1)d: + 1 )d + l )c) = 3x The construction above provides at most l + isolated vertices, K r = fx + c; x + c; x? d + 3c; x + d; x? d + 4c; x + d? c; :::; wg [ f3x? d + c; 3xg where Alternatively, x + l? l?4 w = )d? )c; for l even; x? l?1 l+3 )d + )c; for l odd: K r = 3 +?1)m [ f3x? d + c; 3xg: )x +?1) m d m e? 1)d + 1? d m?1)m e) + 3 ) l )c m=1 5

6 We observe further that by choosing x = lc? l? 1)d, c = 1, and d =?l, we can easily ensure that the construction gives a sum graph whose sum number is at most n +. By Theorem 1 the sum number is at least n + and so we have established Theorem For n even, W n ) = n +. 3 Odd Wheels In this section we disprove the conjecture of Hartseld and Smyth [6] that for odd n 7, W n ) = n+ by showing that for odd n 5, W n ) = n. We begin by noting once again the result [6] that for odd n, every sum graph G = W n +K r is exclusive. This obviously implies that at least n isolated vertices are required for G = W n + K r to be a sum graph. To achieve the minimum sum numbering, we construct the labelling of W n for n odd based on a recent paper [8] that deals with a closely related problem, the integral sum number of cycles. A simple undirected graph G is called an integral sum graph if there exists a labelling of the vertices of G into distinct integers such that any two distinct vertices u and v of G are adjacent if and only if there is a vertex w whose label w) = u) + v). Thus the main dierence between the integral sum graph and the sum graph is that the integral sum graph uses distinct integers for labelling, whereas the sum graph uses distinct positive integers. The integral sum number H) is the least number of isolated vertices K r such that G = H + K r is an integral sum graph. In [8] Sharary showed that the integral sum number of cycles is given by C n ) = 0 if n 6= 4 3 if n = 4. To obtain C n ) = 0 for odd n, the vertices of the cycle C n can be labelled as follows. n = 3 ) V = f?1; 0; 1g; n = 5 ) V = f1; ;?1; 3;?g; n = 7 ) V = f4; 3; 1; ;?5; 7;?3g; n = 9 ) V = f?1;?3;?4; 1;?15; 8;?7; 15;?14g; n = 11 ) V = f?1; 4; 3; 1;?3; 15;?8; 7;?6; 1;?g; n 13 ) V = fb 1 ; b ; :::; b n?; d n?1; d n g; 6

7 where b 1 = 4; b = 1; b i = b i?? b i?1 for i = 3; 4; :::; n? ; d n?1 = b 1 + b? b n? = 5? b n? ; and d n = b n?? b 1 = b n?? 4: Lemma 4 Let a i, i = 1; ; : : : ; n, denote the labels of the vertices under the integer sum labelling that achieves C n ) = 0. Then L = + c where c 3j min n i=1 fa igj+1 is a sum labelling of W n for odd n 5 with n isolated vertices. Proof Let v i, for every i = 1; ; :::; n, be the labels of vertices around the rim of W n and c be the label of the centre of W n. Let v i = 3a i + c, for every i = 1; ; :::; n. Then v i + c = 3a i + c; and v i + v i+1 = 3a i + a i+1 ) + c: Let R be the set of isolated vertices which label the spokes and S be the set of isolated vertices which label the rim. Then R = f3a i + cji = 1; ; :::; ng; and S = f3a i + a i+1 ) + cji = 1; ; :::; ng: For n f5; 7; 9; 11g, we can see that S R and so that S [ R = R. Therefore K n = R = f3a i + cji = 1; ; :::; ng: Similarly, for odd n 13 this construction gives R = f3a i + cji = 1; ; :::; ng: The set S can be obtained in the following way. For i = 3; 4; :::; n?, while Then v i?1 + v i = 3a i? + c v n? + v n?1 = 3a 5 + c; v n?1 + v n = 3a + c; v n + v 1 = 3a n? + c; v 1 + v = 3a 5 + c: S = f3a i + cji = 1; ; :::; n? 4g [ f3a n? + cg: 7

8 Note that S R. Therefore K n = R = f3a i + cji = 1; ; :::; ng: We note that for n = 3 any minimal integral labelling of C 3 must contain 0 as a label which means that using the above construction for W 3 would result in one of the vertices on the rim having the same label as the centre which is not allowed. In fact, W 3 = K4 and so W 3 ) = K 4 ) = 5. We have established Theorem 3 For odd n 5, W n ) = n. Acknowledgements The work of the fourth author was supported by Grant No. A8180 of the Natural Sciences & Engineering Research Council of Canada and by Grant No. GO of the Medical Research Council of Canada. References [1] D. Bergstrand, F. Harary, K. Hodges, G. Jennings, L. Kuklinski & J. Wiener, The sum number of a complete graph, Bull. Malaysian Math. Soc ) 5-8. [] Mark N. Ellingham, Sum graphs from trees, Ars Combinatoria ) [3] R. Gould & V. Rodl, Bounds on the number of isolated vertices in sum graphs, Graph Theory, Combinatorics and Applications edited by Y. Alevi, G. Chartrand, O. R. Oellermann & A. J. Schwenk), John Wiley & Sons 1991) [4] F. Harary, Sum graphs and dierence graphs, Congressus Numerantium ) [5] Nora Hartseld & W. F. Smyth, The sum number of complete bipartite graphs, Graphs and Matrices edited by Rolf Rees), Marcel Dekker 199) [6] Nora Hartseld & W. F. Smyth, A family of sparse graphs of large sum number, Discrete Math )

9 [7] Joseph Ryan, Mirka Miller & W. F. Smyth, The sum labelling for the cocktail party graph, submitted for publication. [8] Ahmad Sharary, Integral sum graphs from complete graphs, cycles and wheels, Arab Gulf J. Sci. Res. 1996). 9