Growth: A Journal of Mathematics and Mathematics Education

ISSN: xxxx-xxxx

Growth: A Journal of Mathematics and Mathematics Education aims to provide a publication platform for high quality undergraduate research in mathematics and in mathematical pedagogy. The technical scope of the journal is combinatorial mathematics, broadly interpreted—the editorial board will consider all submissions in their areas of interest. All submitted articles must have an undergraduate research component and must be certified by a senior researcher. All submissions will be peer reviewed according to standard practices in academic mathematics. Precise editorial policies are set by the editorial board.

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Non-Isomorphic Solutions For Some Triple System with Bipoints

M.J. Grannell1, T.S. Griggs1, K.A.S. Quinn1
1 Department of Pure Mathematics Walton Hall, The Open University Milton Keynes, MK7 6AA United Kingdom
Abstract:

We determine the number of non-isomorphic triple systems with bipoints in those cases for which the total number of triples does not exceed 20.

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Non-Preemptive Scheduling to Maximize the Minimum Intercompletion Time

Carlos C.Amaro1, Sanjoy K.Baruah1, Thomas J.Marlowe2, Alexander D.Stoyen3
1De- partment of Computer & Information Science, New Jersey Institute of Technology, Uni- versity Heights, Newark, NJ 07102 USA
2Department of Math- ematics and Computer Science, Seton Hall University, South Orange, NJ 07079
3Department of Computer Science, The University of Vermont, Burlington, VT 05405
Abstract:

Temporal load-balancing – “spreading out” the executions of tasks over time — is desirable in many applications. A form of temporal load-balancing is introduced: scheduling to maximize minimum inter-completion time (MICT-scheduling). It is shown that MICT-scheduling is, in general, NP-hard. A number of restricted classes of task systems are identified, which can be efficiently MICT-scheduled.

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Counting Configurations of Vectors in a Finite Vector Space with an Orthogonal, Symplectic or Unitary Geometry

M. Q.Rieck
Abstract:

Given a finite-dimensional vector space \(V\) over a finite field \(F\) of odd characteristic, and equipping \(V\) with an orthogonal (symplectic, unitary) geometry, the following two questions are considered:

  1. Given some linearly independent vectors \(w_1, w_2, \ldots, w_k \in V\) and the \(k \times k\) matrix \(A = (\langle w_i, w_j\rangle)\), and given scalars \(\alpha_1, \alpha_2, \ldots, \alpha_k, \beta \in F\), how many vectors \(v \in V\), not in the linear span of \(w_1, w_2, \ldots, w_k\), satisfy \(\langle w_i, v\rangle = \alpha_i\) (\(i = 1, 2, \ldots, k\)) and \(\langle v, v\rangle = \beta\)?
  2. Given a \(k \times k\) matrix \(A = (\lambda_{ij})\) with entries from \(F\), how many \(k\)-tuples \((v_1, v_2, \ldots, v_k)\) of linearly independent vectors from \(V\) satisfy \(\langle v_i, v_j\rangle = \lambda_{ij}\) (\(i, j= 1, 2, \ldots k\))?

      3. An exact answer to the first question is derived. Here there are two cases to consider, depending on whether or not the column vector \((\alpha_i)\) is in the column space of \(A\). This result can then be applied iteratively to address the second question.

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Star Sets in Regular Graphs

Peter Rowlinson1
1Mathematics & Statistics Group Department of Computing Science & Mathematics University of Stirling Scotland, FK9 4LA
Abstract:

Let \(G\) be a finite graph and let \(\mu\) be an eigenvalue of \(G\) of multiplicity \(k\). A star set for \(\mu\) may be characterized as a set \(X\) of \(k\) vertices of \(G\) such that \(\mu\) is not an eigenvalue of \(G – X\). It is shown that if \(G\) is regular then \(G\) is determined by \(\mu\) and \(G – X\) in some cases. The results include characterizations of the Clebsch graph and the Higman-Sims graph.

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Large Neighbourhood Unions and Edge-disjoint Perfect Matchings

Sindi Sabourin1
1Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Canada
Abstract:

We show that if, for any fixed \(r\), the neighbourhood unions of all \(r\)-sets of vertices are large enough, then \(G\) will have many edge-disjoint perfect matchings. In particular, we show that given fixed positive integers \(r\) and \(c\) and a graph \(G\) of even order \(n\), if the minimum degree is at least \(r + c – 1\) and if the neighbourhood union of each \(r\)-set of vertices is at least \(n/2 + \left(2\lfloor\frac{(c + 1)}{2}\rfloor – 1\right)r\), then \(G\) has \(c\) edge-disjoint perfect matchings, for \(n\) large enough. This extends earlier work by Faudree, Gould and Lesniak on neighbourhood unions of pairs of vertices.

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The Fine Structure of Balanced Ternary Designs with Block Size Three, Index Three and \(\rho_2 = 1,2\)

Peter Adams1, Darryn E. Bryant1, A. Khodkar1
1Centre for Discrete Mathematics and Computing Department of Mathematics The University of Queensland Queensland 4072 Australia
Abstract:

In this paper, necessary and sufficient conditions for a vector to be the fine structure of a balanced ternary design with block size \(3\), index \(3\) and \(\rho_2 = 1\) and \(2\) are determined, with one unresolved case.

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Covering the Powers of the Complete Graph with a Bounded Number of Snakes

Salar Y. Alsardary1
1Department of Mathematics, Physics, and Computer Science University of the Sciences in Philadelphia 600 South 43rd Street Philadelphia, PA 19104-4495
Abstract:

Let \(K^d_n\) be the product of \(d\) copies of the complete graph \(K_4\). Wojciechowski [4] proved that for any \(d \geq 2\) the hypercube \(K^d_2\) can be vertex covered with at most \(16\) disjoint snakes. We show that for any odd integer \(n \geq 3\), \(d \geq 2\) the graph \(K^d_n\) can be vertex covered with \(2n^3\) snakes.

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On the Symmetry Groups of Hypergraphs of Perfect Cwatsets

Daniel K. Biss1
1Department of Mathematics Harvard University Cambridge, MA 02138
Abstract:

Cwatsets are subsets of \(\mathbb{Z}^d_2\) which are nearly subgroups and which naturally appear in statistics and coding theory [8]. Each cwatset can be represented by a highly symmetric hypergraph [7]. We introduce and study the symmetry group of the hypergraph and connect it to the corresponding cwatset. We use this connection to establish structure theorems for several classes of cwatsets.

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Cycles Through Specified Vertices in \(1\)-Tough Graphs

Ladislav Stacho 1
1Institute for Informatics, Slovak Academy of Sciences P.O.Box 56, 840 00 Bratislava 4, Slovakia
Abstract:

Bollobás, Brightwell [1] and independently Shi [3] proved the existence of a cycle through all vertices of degree at least \(\frac{n}{2}\) in any \(2\)-connected graph of order \(n\). The aim of this paper is to show that the above degree requirement can be relaxed for \(1\)-tough graphs.

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The \(k\)th Lower Multiexponent of Tournament Matrices

Bolian Liu1, Wang Yan1
1Department of Mathematics South China Normal University Guangzhou,P.R.China
Abstract:

In this paper we investigate the \(k\)th lower multiexponent \(f(n,k)\) for tournament matrices.
It was proved that \(f(m,3) = 2\) if and only if \(m \geq 11\). Thus the conjecture in [2] is disproved. Further we obtain a new sufficient condition for \(f(n,k) = 1\).

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The Combinatorial Press Editorial Office routinely extends invitations to scholars for the guest editing of Special Issues, focusing on topics of interest to the scientific community. We actively encourage proposals from our readers and authors, directly submitted to us, encompassing subjects within their respective fields of expertise. The Editorial Team, in conjunction with the Editor-in-Chief, will supervise the appointment of Guest Editors and scrutinize Special Issue proposals to ensure content relevance and appropriateness for the journal. To propose a Special Issue, kindly complete all required information for submission;
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