Embedding G1-designs into G2-designs, a short survey

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Supported by MIUR, Italy and CNR-GNSAGA
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Definition 1. A G-decomposition of λH (or a (λH, G)-decomposition) is a partition of the edges of λH into subgraphs (G-blocks) each of which is isomorphic to G. A (λH, G)-decomposition is denoted by (V, C ), where V = V (H ) is the vertex set of λH , and C is the G-block-set. A (λKn , G)-decomposition is called a G-design of order n and index λ. A G-design of order v and index λ is called a • path design P (v, k, λ) if G = Pk , the path of length k − 1 (k vertices); • BIBD Sλ (2, k, v ) if G = Kk . An S (2, 3, v ) is well-known as a Steiner triple system. • m-cycle system if G = Cm , the cycle of length m. See [1, 2, 3, 4, 27, 28, 49] for results on (λH, G)-decompositions. Definition 2. We say that a G-design (W, B ) is a subdesign of (V, C ) if W ⊆ V and B ⊆ C . Definition 3. A (λH, G)-decomposition (V, C ) is balanced if each vertex belongs to the same number of blocks. Definition 4. An (almost) G-factor is a spanning subgraph of (λH − v , for some vertex v ) λH such that each component of the (almost) G-factor is isomorphic to G. A (λH, G)-decomposition (V, C ) is (almost) resolvable if C can be partitioned into (almost) G-factors. A (resolvable or almost resolvable) (λKv , Kk )-design is well-known as a (resolvable or almost resolvable) (v, k, λ)-BIBD. Definition 5. Let G1 be a subgraph of G2 and let V and W be two sets such that |V | = v , |W | = w, V ⊆ W . Denote by (V, B ) a G1 -design of order v and index λ1 , and by (W, C ) a G2 -design of order n and index λ2 . • (V, B ) is embedded into (W, C ) if there is a mapping f : B → C such that B is subgraph of f (B ) for every B ∈ B . Equivalentely we say that (V, B ) is embedded into (W, C ) if every G1 block B ∈ B is subgraph of some G2 -block C ∈ C . 2
Embedding G1-designs into G2-designs, a short survey
Gaetano Quattrocchi ∗ Dipartimento di Matematica e Informatica, Universit` a di Catania viale A. Doria, 6 95125 Catania, Italia
• Nested designs [9, 14, 20, 21, 22, 23, 29, 30, 31, 38, 39, 41, 42, 48]. From now on we will consider only strict embedding and we will suppose λ1 = λ2 = 1. So we will write “embedding” instead of “strict embedding”. The following problems arise. Problem 1. For each admissible v , determine the set S1 (v ) of the integers n such that there exists some G2 -design of order n which embeds some G1 -design of order v . Problem 2. For every admissible v , determine the set S2 (v ) ⊆ S1 (v ) such that for every G1 -design (V, B ) of order v and for every n ∈ S2 (v ), there exists at least one G2 -design of order n which embeds (V, B ). Problem 3. For each admissible n and for each integer m ≥ 2, construct a G2 -design (V, C ) of order n such that: 1. it is possible to partition V into m mutually disjoint subsets V1 , V2 , . . . , Vm ; 2. for every i = 1, 2, . . . , m, there exists a G1 -design (Vi , Bi ) embedded into (V, C ). Our aim is to present a short survey of known results.
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Definitions
• H = (V (H ), E (H )) a graph having vertex set V (H ) and edge-set E (H ); • λH the graph H in which every edge has multiplicity λ. • Kn the complete undirected graph on n vertices; • G a subgraph of Kn having nonisolated vertices.
quattrocchi@dmi.unict.it
Abstract A G2 -design (W, C ) on n points embeds a G1 -design (V, B ) on v < n points if every G1 -block B ∈ B is subgraph of some G2 -block C ∈ C . Interesting results have been done when G2 and G1 are connected graphs having respectively k ≤ 4 and k − 1 vertices. We present a short survey of these resictly embedded into (W, C ) if there is an injective mapping f : B → C such that B is subgraph of f (B ) for every B ∈ B . Example 1. Every affine plane of order n is strictly embedded into some projective plane. Example 2. If G1 = G2 and λ1 = λ2 , then we obtain the embedding as in Doyen-Wilson type results [18]. Example 3. A P2 -design (V, B ) of order 3 embedded into a balanced P3 design (W, C ) of order 5: V = {0, 1, 2}, W = {0, 1, . . . , 4}, B = {[0, 1], [1, 2], [0, 2]} and C = {[0, 1, 4], [1, 2, 0], [2, 3, 1], [3, 4, 2], [4, 0, 3]}. Every P2 -design of order v is embedded (but not strictly embedded) into every G-design of order n ≥ v . Example 4. A balanced P3 -design (V, B ) of order 5 strictly embedded into a 4-cycle system (W, C ) of order 9: V = {0, 1, . . . , 4}, W = {0, 1, . . . , 8}, B = {[0, 4, 1], [2, 0, 3], [0, 1, 2], [4, 2, 3], [1, 3, 4]} and C = {(0, 4, 1, 6), (2, 0, 3, 7), (0, 1, 2, 5), (4, 2, 3, 6), (1, 3, 4, 5), (7, 0, 8, 1), (6, 2, 8, 5), (5, 3, 8, 7), (7, 4, 8, 6)}. The embedding definition is quite general. Different “classes of designs” can be seen as a G-design (V, C ) of order v and index λ such that: 1. for every i = 1, 2, . . . , p, (W, C ) embeds a Gi -design (V, Bi ) of order v and index λi ; 2. λ1 + λ2 + . . . + λp = λ; and in some cases 3. the embedding fi : Bi → C is a bijective mapping. As examples of these “classes of designs” we cite: • Edge-colored-designs [15, 6, 7, 8, 32]. • Complementary decomposition [19, 25, 46, 47]. • Steiner pentagon system [34, 35]. 3