Self-orthogonal decompositions of graphs into matchings

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<ul><li><p>Self-orthogonal decompositions of graphs intomatchings</p><p>Sven Hartmann 1</p><p>Dept. of Information SystemsMassey University</p><p>Palmerston North, New Zealand</p><p>Uwe Leck 2</p><p>Institut fur MathematikUniversitat RostockRostock, Germany</p><p>Abstract</p><p>Given a simple graph H, a self-orthogonal decomposition (SOD) of H is a collectionof subgraphs of H, all isomorphic to some graph G, such that every edge of H occursin exactly two of the subgraphs and any two of the subgraphs share exactly one edge.Our concept of SOD is a natural generalization of the well-studied orthogonal doublecovers (ODC) of complete graphs. If for some given G there is an appropriate H,then our goal is to nd one with as few vertices as possible. Special attention is paidto the case when G a matching with n1 edges. We conjecture that v(H) = 2n2is best possible if n = 4 is even and v(H) = 2n if n is odd. We present a constructionwhich proves this conjecture for all but 4 of the possible residue classes of n modulo18.</p><p>Keywords: graph decomposition, factorization, self-orthogonal factorization,orthogonal double cover, ODC</p><p>Electronic Notes in Discrete Mathematics 23 (2005) 511</p><p>1571-0653/$ see front matter 2005 Elsevier B.V. All rights reserved.</p><p></p><p>doi:10.1016/j.endm.2005.06.102</p></li><li><p>1 Introduction</p><p>1.1 Orthogonal double covers of complete graphs</p><p>The concept of orthogonal double covers of complete graphs originates inpapers by Hering and Rosenfeld [7] on a question from statistical design ofexperiments and by Demetrovics, Furedi and Katona [4] who studied Arm-strong databases of minimum size for key and functional dependencies. Sincethen, dozens of research papers on the subject have been published; for anoverview we refer to the survey [5].</p><p>An orthogonal double cover (ODC) of the complete graph Kn is a collectionG = {Gi : i = 1, 2, . . . , n} of spanning subgraphs of Kn, called pages, such thatthe following two conditions are satised:</p><p>(i) Double cover conditionEvery edge of Kn belongs to the edge set of exactly two pages of G.</p><p>(ii) Orthogonality conditionAny two distinct pages of G share exactly one edge.</p><p>The above denition immediately implies that every page of G must haveexactly n 1 edges. If all pages of G are isomorphic to some graph G, then Gis said to be an ODC of Kn by G.</p><p>The problem usually studied in this context is: Given a class C of graphs,decide for which G C there is an ODC of the corresponding complete graphby G. For instance, the problem in [7] turned out to be equivalent to theexistence of ODC of complete graphs by almost-hamiltonian cycles, Figure 1shows a solution for n = 5.</p><p>0</p><p>23</p><p>3</p><p>1</p><p>0</p><p>23</p><p>1</p><p>0</p><p>23</p><p>1</p><p>0</p><p>23</p><p>1</p><p>0</p><p>23</p><p>14 4 4 4 4</p><p>G G G G G1 2 40Fig. 1. An ODC of K5 by C4 {v}</p><p>It is very natural to ask the above question for C being the class of all trees1 Email: Email:</p><p>S. Hartmann, U. Leck / Electronic Notes in Discrete Mathematics 23 (2005) 5116</p></li><li><p>as e(G) = n1 is necessary for the existence of an ODC of Kn by G. One caneasily observe that there is no ODC of K4 by P4, the path of length three. Forall other non-trivial trees on at most 14 vertices ODC of the correspondingcomplete graphs exist [5] which supports the following conjecture.</p><p>Conjecture 1.1 ([6]) If T = P4 is a tree on n 2 vertices, then there is anODC of Kn by T .</p><p>1.2 Self-orthogonal graph decompositions</p><p>We now introduce a straight forward generalization of ODC of completegraphs. For graphs G which do not admit an ODC of some complete graphthis will provide a way to measure how close to such an ODC one can get.Furthermore, our generalized concept will allow to include also graphs G thatdo not satisfy the necessary condition e(G) = n 1.</p><p>Let G be a graph with e(G) = n 1. Clearly, an ODC of Kn by G canonly exist if v(G) = n. If v(G) &lt; n, then we could simply add n v(G)isolated vertices to G and ask for an ODC of Kn by the resulting graph. Ifv(G) &gt; n, for instance if G is a forest with more than one component, this isnot possible. This simple observation motivates the following denition.</p><p>A self-orthogonal G-decomposition (G-SOD) of a graph H is a collectionG = {Gi : i = 1, 2, . . . , n} of subgraphs of H such that:(i) Every edge of H belongs to exactly two members of G.(ii) Any two distinct graphs from G share exactly one edge.(iii) Gi is isomorphic to G for i = 1, 2, . . . , n.</p><p>Note that the term ODC of H is already used for a slightly dierent gen-eralization of ODC of complete graphs (see [5]). In order to avoid confusionwith this concept, we changed the terminology signicantly.</p><p>As e(G) = n 1, it follows immediately that a necessary condition for anG-SOD of H to exist is e(H) =</p><p>(n2</p><p>). Given G, a natural question arising is</p><p>to nd GSOD that are as compact as possible, that is, to minimize v(H)over all graphs H that allow a G-SOD of H. If for some graphs G there is noappropriate H, then put (G) := , otherwise we dene</p><p>(G) := min{v(H) : There is a G-SOD of H.}</p><p>Clearly, (G) n with equality if and only if there is an ODC of Kn by G.Conjecture 1.1 can now be restated in terms of SOD as follows: If T is</p><p>a tree on n 2 vertices, then (T ) = n. As mentioned before, there is no</p><p>S. Hartmann, U. Leck / Electronic Notes in Discrete Mathematics 23 (2005) 511 7</p></li><li><p>ODC of K4 by P4, so (P4) &gt; 4. Figure 2 shows a P4-SOD of a graph on vevertices, so (P4) = 5. (An edge labelled ij means that this edge is containedin the i-th and j-th copies of P4.)</p><p>12</p><p>13</p><p>24</p><p>23</p><p>34</p><p>14</p><p>Fig. 2. An SOD of a graph into four paths of length 3</p><p>Note that there are innitely many graphs with (G) = : If G is a cliqueon m vertices, then a G-SOD is equivalent to a biplane with block size m (see[3] for denition). Such biplanes are known to not exist for innitely manyvalues of m due to the Bruck-Ryser-Chowla Theorem [1,2].</p><p>2 SOD of graphs into matchings</p><p>For n 2, let Mn denote the graph consisting of n 1 independent edges.We study the problem of determining (Mn).</p><p>2.1 Preliminary observations</p><p>Obviously, (Mn) v(Mn) = 2n2 holds. On the other hand, one can easilyconstruct an Mn-SOD of Mk with k =</p><p>(n2</p><p>)+ 1. This implies that (Mn) is</p><p>nite for all n and gives rst trivial bounds.</p><p>Proposition 2.1 2n 2 (Mn) n2 n holds for all n 2.For some n 2, let {Gi : i = 1, 2, . . . , n + 1} be an Mn+1-SOD of some</p><p>graph H. Let Gi be obtained from Gi removing the common edge of Gi andGn+1, i = 1, 2, . . . , n. Then {Gi : i = 1, 2, . . . , n} is an Mn-SOD. By this littleobservation we have:</p><p>Proposition 2.2 (Mn+1) (Mn) holds for all n 2.For small n one can easily establish the following values of (Mn):</p><p>n 2 3 4 5 6 7 8</p><p>(Mn) 2 6 9 10 10 14 14</p><p>S. Hartmann, U. Leck / Electronic Notes in Discrete Mathematics 23 (2005) 5118</p></li><li><p>A corresponding SOD for n = 6 is displayed in Figure 3. It turns out that(Mn) for small n is close to the lower bound in Proposition 2.1.</p><p>13</p><p>56</p><p>16</p><p>35</p><p>26</p><p>14</p><p>36</p><p>25</p><p>46</p><p>12</p><p>23</p><p>34</p><p>45</p><p>15</p><p>24</p><p>Fig. 3. An M6SOD of the Petersen graph</p><p>Nevertheless, the bound cannot be tight for odd n:</p><p>Lemma 2.3 If n 3 is odd, then (Mn) 2n.Proof. Let n 3 be odd, and let G = {Gi : i = 1, 2, . . . , n} be an Mn-SOD of H. Consider some x V (H). As every edge of H occurs in exactlytwo members of G, the vertex x is contained in an even number of graphsfrom G, i.e. in at most n 1. Every edge from H that contains x is inexactly two graphs from G, and therefore dH(x) (n 1)/2. Summing upover all x V (H) gives 2e(H) (n 1)v(H)/2, and v(H) 2n follows bye(H) =</p><p>(n2</p><p>). </p><p>Obviously, the argument in the proof of Lemma 2.3 can also be used toshow:</p><p>Lemma 2.4 Let n 2. If H is a graph with</p><p>v(H) =</p><p>2n 2 if n is even,2n if n is odd</p><p>such that there is an Mn-SOD of H, then H is n/2-regular.We conjecture that such a graph H exists for all n = 4.</p><p>S. Hartmann, U. Leck / Electronic Notes in Discrete Mathematics 23 (2005) 511 9</p></li><li><p>Conjecture 2.5 If 2 n = 4, then</p><p>(Mn) =</p><p>2n 2 if n is even,2n if n is odd.</p><p>By Proposition 2.2, it would be sucient to prove the above conjecture forall even n.</p><p>2.2 Construction and main result</p><p>For simplicity, assume for the moment that n 1 = q 5 is a prime powerwith q 1 (mod 4). Let be a primitive element of GF(q). On the vertexset V = V (H) = GF(q) Z2, we dene maximal matchings Gx, x GF(q),and G by</p><p>E(Gx)={{(2i + x, 0), (2i+1 + x, 0)} : i = 0, 1, . . . , (q 3)/2</p><p>}</p><p>{{(2i+1 + x, 1), (2i+2 + x, 1)} : i = 0, 1, . . . , (q 3)/2</p><p>}</p><p>{{(x, 0), (x, 1)}</p><p>}</p><p>and</p><p>E(G) ={{(x, 0), (x, 1)} : x GF(q)</p><p>},</p><p>respectively. One can check that the collection</p><p>G := {Gx : x GF(q)} {G}</p><p>is an Mn-SOD of H :=</p><p>xGF(q) Gx. Consequently, (Mn) = (Mn1) =2n 2 holds. The SOD in Figure 3 has been generated applying the aboveconstruction to n = 6.</p><p>If n 1 = q = 3 is a prime power with q 3 (mod 4), then a similarconstruction turns out to work, where this time instead of a primitive elementwe work with an element of order (q 1)/2.</p><p>A common generalization of both constructions will be given in the fullversion of this paper. This general form of our construction can be appliedto all even n with prime factorization n 1 = mi=1 pkii such that pkii = 3 forall i. It uses the action of Z2 Zk1p1 Zkmpm to generate an Mn-SOD of agraph H with v(H) = 2n 2. Therefore, the following theorem holds.</p><p>S. Hartmann, U. Leck / Electronic Notes in Discrete Mathematics 23 (2005) 51110</p></li><li><p>Theorem 2.6 Let n 2 be a natural number with n 3, 4, 15, 16 (mod 18).Then</p><p>(Mn) =</p><p>2n 2 if n is even,2n if n is odd.</p><p>The full version will also contain more sophisticated constructions for someof the missing cases, but so far these only give solutions for a class of valuesfor n whose arithmetic density is 0.</p><p>References</p><p>[1] Bruck, R.H., and H.J. Ryser, The nonexistence of certain nite projectiveplanes, Canad. J. Math. 1 (1949), 8893.</p><p>[2] Chowla, S., A property of biquadratic residues, Proc. Nat. Acad. Sci. IndiaA 14 (1944), 4546.</p><p>[3] Colbourn, C.J., and J.H. Dinitz (eds.), The CRC Handbook ofCombinatorial Designs, CRC Press, Boca Raton, FL, 1996.</p><p>[4] Demetrovics, J., Z. Furedi, and G.O.H. Katona, Minimum matrixrepresentations of closure operations, Discrete Appl. Math. 11 (1985), 115128.</p><p>[5] Gronau, H.-D.O.F., M. Gruttmuller, S. Hartmann, U. Leck, and V. Leck,On orthogonal double covers of graphs, Des. Codes Cryptogr. 27 (2002),4991.</p><p>[6] Gronau, H.-D.O.F., R.C. Mullin, and A. Rosa, Orthogonal double covers ofcomplete graphs by trees, Graphs Combin. 13 (1997), 251262.</p><p>[7] Hering, F., and M. Rosenfeld, Problem Number 38, In (K. Heinrich,ed.): Unsolved Problems: Summer Research Workshop in AlgebraicCombinatorics, Simon Fraser University, 1979.</p><p>S. Hartmann, U. Leck / Electronic Notes in Discrete Mathematics 23 (2005) 511 11</p></li></ul>