On a Problem in Combinations

Electronic notes in discrete mathematics

A complete solution to the two-table Oberwolfach problems

In 1980, about 20 years after sequenceable groups were introduced by Gordon to construct row-complete latin squares, Keedwell published a survey of all the available results concerning sequencings. This was updated (jointly with Denes) in 1991 and a short overview, including results about complete mappings and R-sequencings, was given in the CRC Handbook of Combinatorial Designs in 1995. In Sections 1 and 2 we give a survey of the current situation concerning sequencings, including details of the most important constructions. In Section 3 we consider some concepts closely related to sequenceable groups: R-sequencings, harmonious groups, supersequenceable groups (also known as super P-groups), terraces and the Gordon game. We also look at constructions for row-complete latin squares that do not use sequencings.

/pdf/sequenceable-groups-and-related-topics-4d6xmg4hm8.pdf

Sequenceable Groups and Related Topics

The urban ecological environment is the material basis and condition for human beings to engage in social and economic activities and the supporting system for the formation and sustainable development of cities. With the acceleration of urbanization and industrialization, urban living environments and economic development have become the focus of people’s attention. This leads to the necessity of studying how to improve the quality of the urban living environment and promote the harmonious coexistence of population, natural environment, and social economy. Traditional methods focus on multiple regression models to evaluate the urban environmental and economic harmony, but this method does not consider the weight of each index, resulting in poor accuracy of the evaluation results. This paper proposes a discrete mathematical model to design the evaluation index and evaluation system of urban environmental and economic coordination. It calculates the weight of each index; carrying capacity of the urban environment, the value of each environmental factor, and the comprehensive value of the environment is determined. The static evaluation and dynamic evaluation are used to evaluate the coordination of the urban environmental economy. The experimental results show that the designed evaluation method of urban environmental economic coordination has high accuracy and effectively improves the reliability and evaluation time.

/pdf/evaluation-of-urban-environmental-and-economic-coordination-373e62f71x.pdf

Evaluation of Urban Environmental and Economic Coordination Based on Discrete Mathematical Model

Every 1-rotational solution of a classic or twofold Oberwolfach problem (OP) of order n is generated by a suitable 2-factor (starter )o fKn or 2Kn, respectively. It is shown that any starter of a twofold OP of order n gives rise to a starter of a classic OP of order 2n − 1 (doubling construction). It is also shown that by suitably modifying the starter of a classic OP, one may obtain starters of some other OPs of the same order but having different parameters. The combination of these two constructions leads to lots of new infinite classes of solvable OPs. Still more classes can be obtained with the help of a third construction making use of the possible gracefulness of a graph whose connected components are cycles and at most one path. As one of the many applications, Hilton and Johnson's (J London Math Soc, 64 (2001) 513-522) bound s ≥ 5r − 1 about the solvability of OP(r, s) is improved to s ≥� r/4 �+ 10 in the case of r even. C � 2012 Wiley Periodicals, Inc. J. Combin. Designs 20: 483-503, 2012

2-Starters, Graceful Labelings, and a Doubling Construction for the Oberwolfach Problem

Let Kv∗ denote the complete graph Kv if v is odd and Kv−I, the complete graph with the edges of a 1-factor removed, if v is even. Given nonnegative integers v,M,N,α,β, the Hamilton–Waterloo problem asks for a 2-factorization of Kv∗ into α CM-factors and β CN-factors, with a Cl-factor of Kv∗ being a spanning 2-regular subgraph whose components are l-cycles. Clearly, M,N≥3, M∣v, N∣v and α+β=⌊v−12⌋ are necessary conditions. In this paper, we extend a previous result by the same authors and show that for any odd v≠MNgcd(M,N) the above necessary conditions are sufficient, except possibly when α=1, or when β∈{1,3}. Note that in the case where v is odd, M and N must be odd. If M and N are odd but v is even, we also show sufficiency but with further possible exceptions. In addition, we provide results on 2-factorizations of the complete equipartite graph and the lexicographic product of a cycle with the empty graph.

On the Hamilton–Waterloo problem with odd cycle lengths

A Hamiltonian cycle system of Kv (briefly, a HCS(v)) is 1-rotational under a (nec- essarily binary )g roupG if it admits G as an automorphism group acting sharply transitively on all but one vertex. We first prove that for any n ≥ 3 there exists a 3-perfect 1-rotational HCS(2n + 1). This allows to get the existence of another infinite class of 3-perfect (but not Hamiltonian) cycle decompositions of the complete graph. Then we prove that the full auto- morphism group of a 1-rotational HCS under G is G itself unless the HCS is the 2-transitive one. This allows us to give a partial answer to the problem of determining which abstract groups are the full automorphism group of a HCS. Finally, we revisit and simplify by means of a careful group theoretic discussion a formula by Bailey, Ollis, and Preece on the number of inequivalent 1-rotational HCSs under G. This leads us to a formula counting all 1-rotational HCSs up to isomorphism. Though this formula heavily depends on some parameters that are hard to compute, it allows us to say that, for any n ≥ 6, there are at least 2 � 3n/4� nonisomorphic

Some Results on 1-Rotational Hamiltonian Cycle Systems

A Steiner triple system of order v (briefly STS(v)) is 1-rotational under G if it admits G as an automorphism group acting sharply transitively on all but one point. The spectrum of values of v for which there exists a 1-rotational STS(v) under a cyclic, an abelian, or a dicyclic group, has been established in Phelps and Rosa (Discrete Math 33:57---66, 1981), Buratti (J Combin Des 9:215---226, 2001) and Mishima (Discrete Math 308:2617---2619, 2008), respectively. Nevertheless, the spectrum of values of v for which there exists a 1-rotational STS(v) under an arbitrary group has not been completely determined yet. This paper is a considerable step forward to the solution of this problem. In fact, we leave as uncertain cases only those for which we have v = (p 3?p)n + 1 ? 1 (mod 96) with p a prime, $${n \not\equiv 0}$$ (mod 4), and the odd part of (p 3 ? p)n that is square-free and without prime factors congruent to 1 (mod 6).

pdf/some-progress-on-the-existence-of-1-rotational-steiner-2dikrz5ld0.pdf

Some progress on the existence of 1-rotational Steiner triple systems

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