511 research outputs found
Partial triple systems and edge colourings
AbstractA partial triple system of order v, PT(v), is pair (V, B) where V is a v-set, and B is a collection of 3-subsets of V (called triples) such that each 2-subset of V is contained in at most one triple. A maximum partial triple system of order v, MPT(v), is a PT(v), (V, B), such that for any other PT(v), (V, C), we have |C| ⪕|B|. Several authors have considered the problem of embedding PT(v) and MPT(v) in systems of higher order. We complete the proof, begun by Mendelsohn and Rosa [6], that an MPT(u) can be embedded in an MPT(v) where v is the smallest value in each congruence class mod 6 with v ⩾ 2u. We also consider a general problem concerning transversals of minimum edge-colourings of the complete graph
A Penrose polynomial for embedded graphs
We extend the Penrose polynomial, originally defined only for plane graphs,
to graphs embedded in arbitrary surfaces. Considering this Penrose polynomial
of embedded graphs leads to new identities and relations for the Penrose
polynomial which can not be realized within the class of plane graphs. In
particular, by exploiting connections with the transition polynomial and the
ribbon group action, we find a deletion-contraction-type relation for the
Penrose polynomial. We relate the Penrose polynomial of an orientable
checkerboard colourable graph to the circuit partition polynomial of its medial
graph and use this to find new combinatorial interpretations of the Penrose
polynomial. We also show that the Penrose polynomial of a plane graph G can be
expressed as a sum of chromatic polynomials of twisted duals of G. This allows
us to obtain a new reformulation of the Four Colour Theorem
Computing the chromatic number of t-(v,k,[lambda]) designs
Colouring t-designs has previously been shown to be an NP-complete problem; heuristics and a practical algorithm for this problem were developed for this thesis; the algorithm was then employed to find the chromatic numbers of the sixteen non- isomorphic 2-(25, 4, 1) designs and the four cyclic 2-(19, 3, 1) designs. This thesis additionally examines the existing literature on colouring and finding chromatic numbers of t-designs
New Examples of Kochen-Specker Type Configurations on Three Qubits
A new example of a saturated Kochen-Specker (KS) type configuration of 64
rays in 8-dimensional space (the Hilbert space of a triple of qubits) is
constructed. It is proven that this configuration has a tropical dimension 6
and that it contains a critical subconfiguration of 36 rays. A natural
multicolored generalisation of the Kochen-Specker theory is given based on a
concept of an entropy of a saturated configuration of rays.Comment: 24 page
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