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Decomposition of Complete Graphs Into Stars

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Decomposition of Complete Graphs Into Stars BULL. AUSTRAL. MATH. SOC. 05CI0, O5C99 VOL. 10 (1974), 23-30. Decomposition of complete graphs into stars Pauline Cain A star is a connected graph in which every vertex but one has valency 1 . This paper concerns the question of when complete graphs can be decomposed into stars', all of the same order, which have pairwise disjoint edge-sets. It is shown that the complete graphs on rm and rm + 1 vertices, r > 1 , can be decomposed into stars with m edges, if and only if r- is even or m is odd. By a graph we shall mean a finite undirected graph without loops or multiple edges. In the complete- graph K there are p vertices and an edge exists between every pair of vertices. The complete bipartite graph, K , has two sets of vertices, V and V , and two vertices are p ,71 p n adjacent if and only if both endpoints do not belong to V or to V An m-star is a complete-bipartite graph, K . We shall write x - yztu ... for a star with centre x and terminal vertices y, z, t, u, ... A decomposition or factorization of a graph into stars is a way of expressing the graph as the union of edge-disjoint stars. A uniform decomposition is one in which the stars are the same size. An m-star decomposition decomposes a graph uniformly into m-stars. The sum of graphs, G + H , consists of the union of the vertices and edges in G and H and all possible edges between every pair of vertices Received 20 August 1973. Communicated by W.D. Wallis. 23 Downloaded from https://www.cambridge.org/core. IP address: 170.106.40.139, on 29 Sep 2021 at 08:57:09, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972700040582 24 PauIi ne Ca i n g and h where g belongs to G and h belongs to H . Examples of a l»-star and of a U-star decomposition of X- are given in Figure 1. x - tzyu Figure 1 In [J] Ae, Yamamoto, Yoshlda have shown that K . , for t greater than one, is 3-star decomposable. In this paper we shall show precisely when K is ro-star decomposable. LEMMA 1. If K is m-star decomposable then necessarily the manber of stars, p(p-l)/2m } is integral. LEMMA 2. K is m-star decomposable for all m .. Proof. £_ is 1-star decomposable. (All graphs are trivially 1-star decomposable. ) K\. is 2-star decomposable as is shown in Figure 2. Assume K is p-star decomposable for all p 5 m may be formed from a o K- Joining by edges, e. , each pair of vertices [v., w .) , where v. , for i = 1, 2 , belongs to the and W • , for 3 3 = 1, .... 2m , belongs to the K . (See Figure 3.) Downloaded from https://www.cambridge.org/core. IP address: 170.106.40.139, on 29 Sep 2021 at 08:57:09, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972700040582 Decomposition of graphs 25 Figure 2 3 = 1» 2 2m Figure 3 There are 2m - 1 m-staxs in Kn . Since each vertex in Ko has a 2m 2m valency of 2m - 1 , every vertex (of X? J except one is the centre of an m-star. Label the exception wr. 2m For j = 1, ..., m , attach to the star with centre w. , the edge 3 e . to form an (m+l)-star. Then form an (m+l)-star, Downloaded from https://www.cambridge.org/core. IP address: 170.106.40.139, on 29 Sep 2021 at 08:57:09, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972700040582 26 PaulIne Cain V -U , w , ..., W , u , exhausting the remaining edges through v, . xn do m+± dm—i. dm 1 For o - m+1 2m-l , add e2 . to the m-star with centre W . , forming an (m+l)-star. The remaining (m+l)-star has v as its centre and terminal vertices w, u , u_ . We have a total of 2m + 1 1 m 2m edge-disjoint (m+l)-stars. So the result follows by induction. THEOREM 1. If K is m-star decomposable then K „ for all x x+dam positive integral a j is m-star decomposable. Proof. K Consider K + K . Let V- be the vertices of K for i = 1, ..., x and W. be the I x ' 3 vertices of K for j = 1, .... 2m . Let e. be an edge between v. and w . 3 Both K and K are m-star decomposable by assumption and Lemma 2 respectively. For i = 1, ..., x , the graph, E. , containing the edges e. where g = 1, ..., 2m , is decomposable into two m-stars, namely, 10 V. - W , .... w and v. - w , , ... &?„ . So K + K is decomposable i l m t m+1 2m x 2nm for K and every K . The result follows by repeated application. LEMMA 3. K and K , are not m-star decomposabler when r is im rm+1 odd and m is even. Proof. Lemma 1 implies that m will divide rm(rm-l)/2 and (rm+l)rm/2 if K and K are m-star decomposable. This does not XWl 2v/f+l happen in the case in question. LEMMA 4. K is m-star decomposable when m is odd. Proof. Write m = 2M + 1 . If X_ is decomposable the number of Downloaded from https://www.cambridge.org/core. IP address: 170.106.40.139, on 29 Sep 2021 at 08:57:09, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972700040582 Decomposition of graphs 27 m-stars will be 3(2w+l)(6n+3-l)/2(2n+l) which is an integer, 3(3n+l) . So, for odd m it is conceivable that K__ may be m-star decomposable. let us write these K with vertices V. , trt tn TTI m *• i = 1, ..., m , w. , 3 = 1, ..., m and x, , k = 1, ..., m respectively. (See Figure !». ) i = 1, 2, ... , m j = 1, 2, .. ., m k = 1, 2, ... , n Figure 4 For 1 £ i £ m-1 , form the m-star V. with centre v. and terminal vertices w. , ..., W., V . ,,..., W ; 1 i. z+i m for 2 £ i £ m , form the m-star X. with centre x. and terminal vertices x, ...,x. ,u., ...,u ; for 1 £ i £ m , form the m-star W. with centre w. and terminal vertices U, , . ... V. ., V , x. ,, ..., x ; 1 t—1 m i+i m for 1 £ i £ m , form the m-star V. with centre v. and terminal vertices x , . .., x We have formed km - 2 m-stars and we require a further %(m+l) stars to be formed. We proceed as follows: for 1 £ i £ m , the star v. is centred at w. and has terminal vertices w . , .. , w, x . Y. requires a further t - 1 edges to become an m-star. F, has centre u, and terminal vertices w , . .. , w , x, . So y, 1 1 2n m 1 1 Downloaded from https://www.cambridge.org/core. IP address: 170.106.40.139, on 29 Sep 2021 at 08:57:09, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972700040582 28 Pauline Cain is an m-star already. v In order to form a further %(m-l) m-stars X, where k = 2, ..., %(m+l) , we may use the following procedure to modify the V h °»* h •• (1) For 2< h %(m+l) : to Yk add vk ; to Vk delete ufc and add Wm_k+2 ; to W . delete u, and add x . X will now have m terminal vertices, namely, W3 Wm> XV V2 • (2) (i) If k is less than %(n?+5-j) , write 8=m-fc + 5-J and do the following for 35j'5 %(m+l) and j ± k ± %(m+l) to rfc add »9; to Va add W . _ and delete tO (ii) If k is greater than or equal to %(m+5-j) , write j = m + 5 _ 2k and 6=m-k + 5-j. Do the following for 0 < I < k-j : to *fc add xez ; to XQI add x - and delete w. ; to jr^j add Vj+3-z "* delete delete Vl ^ add Using this procedure X. gains one edge from (l) and (t-2) edges from (2), thus forming an m-star X. for i = 2, ..., %(ml) . J^, and V. are still m-stars since an edge is always added when one is If subtracted. The following sitaations need to be considered to ensure that the Downloaded from https://www.cambridge.org/core. IP address: 170.106.40.139, on 29 Sep 2021 at 08:57:09, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972700040582 Decomposition of graphs 29 procedure does not break down for large m . In (l) the procedure may break down if m-k+2 £ k , but this implies that m+2 £ 2k . Since 2 £ k £ %(m+l) , we have m+2 £ m+1 , which is false. In (2) (i) the procedure may break down if m-fc+5-J £ k > that is k S %(w+5-j) » but this situation is corrected in (2) (ii).
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