Minimum embedding of any Steiner triple system into a 3-sun system via matchings (original) (raw)
Related papers
Equitable embeddings of steiner triple systems
Journal of Combinatorial Theory, Series A, 1996
Using an embedding result for pairwise balanced designs, and colourings of small systems, tripling constructions are used to produce equitably eoloured Steiner triple systems. It is shown that when the order is v is large enough with respect to the number r of colours, and v= 1, 3 (mod6), an equitably r-coloured r-chromatic Steiner triple system of order v exists.
Embedding Partial Steiner Triple Systems
Proceedings of the London Mathematical Society, 1980
We prove that a partial Steiner triple system 8 of order n can be embedded in a Steiner triple system T of any given admissible order greater than 4w. Furthermore, if G(S), the missing-edge graph of S, has the property that A(G)<ri(n + l)l and \E(G)\ then # can be embedded in a Steiner triple system of order 2n +1, provided that 2w +1 is admissible.
On determining when small embeddings of partial Steiner triple systems exist
Journal of Combinatorial Designs, 2020
A partial Steiner triple system of order u is a pair (U, A) where U is a set of u elements and A is a set of triples of elements of U such that any two elements of U occur together in at most one triple. If each pair of elements occur together in exactly one triple it is a Steiner triple system. An embedding of a partial Steiner triple system (U, A) is a (complete) Steiner triple system (V, B) such that U ⊆ V and A ⊆ B. For a given partial Steiner triple system of order u it is known that an embedding of order v 2u + 1 exists whenever v satisfies the obvious necessary conditions. Determining whether "small" embeddings of order v < 2u + 1 exist is a more difficult task. Here we extend a result of Colbourn on the NP-completeness of these problems. We also exhibit a family of counterexamples to a conjecture concerning when small embeddings exist.
Colouring Cubic Graphs by Small Steiner Triple Systems
Graphs and Combinatorics, 2007
Given a Steiner triple system S, we say that a cubic graph G is S-colourable if its edges can be coloured by points of S in such way that the colours of any three edges meeting at a vertex form a triple of S. We prove that there is Steiner triple system U of order 21 which is universal in the sense that every simple cubic graph is U-colourable. This improves the result of Grannell et al. [J. Graph Theory 46 (2004), 15-24] who found a similar system of order 381. On the other hand, it is known that any universal Steiner triple system must have order at least 13, and it has been conjectured that this bound is sharp (Holroyd andŠkoviera [J. Combin. Theory Ser. B 91 (2004), 57-66]).
A Steiner triple system which colors all cubic graphs
Journal of Graph Theory, 2004
We prove that there is a Steiner triple system T such that every simple cubic graph can have its edges coloured by points of T in such a way that for each vertex the colours of the three incident edges form a triple in T. This result complements the result of Holroyd anď Skoviera that every bridgeless cubic graph admits a similar colouring by any Steiner triple system of order greater than 3. The Steiner triple system employed in our proof has order 381 and is probably not the smallest possible.
Colouring of cubic graphs by Steiner triple systems
Journal of Combinatorial Theory, Series B, 2004
Let S be a Steiner triple system and G a cubic graph. We say that G is S-colourable if its edges can be coloured so that at each vertex the incident colours form a triple of S. We show that if S is a projective system P G(n, 2), n ≥ 2, then G is S-colourable if and only if it is bridgeless, and that every bridgeless cubic graph has an S-colouring for every Steiner triple system of order greater than 3. We establish a condition on a cubic graph with a bridge which ensures that it fails to have an S-colouring if S is an affine system, and we conjecture that this is the only obstruction to colouring any cubic graph with any non-projective system of order greater than 3.
Strict colorings of Steiner triple and quadruple systems: a survey
Discrete Mathematics, 2003
The paper surveys problems, results and methods concerning the coloring of Steiner Triple and Quadruple Systems viewed as mixed hypergraphs. In this setting, two types of conditions are considered: each block of the Steiner system in question has to contain (i) a monochromatic pair of vertices, or, more restrictively, (ii) a triple of vertices that meets precisely two color classes.
Journal of Combinatorial Designs, 2005
In this paper, we present a conjecture that is a common generalization of the Doyen-Wilson Theorem and Lindner and Rosa's intersection theorem for Steiner triple systems. Given u, v v 1, 3 ðmod 6Þ, u < v v < 2u þ 1, we ask for the minimum r such that there exists a Steiner triple system ðU, BÞ, jUj ¼ u such that some partial system ðU, Bnq qÞ can be completed to an STSðv vÞ, ðV, B 0 Þ, where jq qj ¼ r. In other words, in order to ''quasi-embed'' an STSðuÞ into an STSðv vÞ, we must remove r blocks from the small system, and this r is the least such with this property. One can also view the quantity ðuðu À 1Þ=6Þ À r as the maximum intersection of an STSðuÞ and an STSðv vÞ with u < v v. We conjecture that the necessary minimum r ¼ ðv v À uÞð2u þ 1 À v vÞ=6 can be achieved, except when u ¼ 6t þ 1 and v v ¼ 6t þ 3, in which case it is r ¼ 3t for t 6 ¼ 2, or r ¼ 7 when t ¼ 2. Using small examples and recursion, we solve the cases v v À u ¼ 2 and 4, asymptotically solve the cases v v À u ¼ 6, 8, and 10, and further show for given v v À u > 2 that an asymptotic solution exists if solutions exist for a run of consecutive values of u (whose required length is no more than v v À u). Some results are obtained for v v close to 2u þ 1 as well. The cases where v v % 3u=2 seem to be the hardest.
Silver Block Intersection Graphs of Steiner 2-Designs
Graphs and Combinatorics, 2012
For a block design D, a series of block intersection graphs G i , or i-BIG(D), i = 0,. .. , k is defined in which the vertices are the blocks of D, with two vertices adjacent if and only if the corresponding blocks intersect in exactly i elements. A silver graph G is defined with respect to a maximum independent set of G, called an α-set. Let G be an r-regular graph and c be a proper (r + 1)-coloring of G. A vertex x in G is said to be rainbow with respect to c if every color appears in the closed neighborhood N [x] = N (x) ∪ {x}. Given an α-set I of G, a coloring c is said to be silver with respect to I if every x ∈ I is rainbow with respect to c. We say G is silver if it admits a silver coloring with respect to some I. Finding silver graphs is of interest, for a motivation and progress in silver graphs see [7] and [15]. We investigate conditions for 0-BIG(D) and 1-BIG(D) of Steiner 2-designs D = S(2, k, v) to be silver.