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.
- Research article
- Full Text
- Ars Combinatoria
- Volume 091
- Pages: 97-111
- Published: 30/04/2009
Let \(T\) be a partially ordered set whose Hasse diagram is a binary tree and let \(T\) possess a unique maximal element \(1_T\). For a natural number \(n\), we compare the number \(A_T^n\) of those chains of length \(n\) in \(T\) that contain \(1_T\) and the number \(B_T^n\) of those chains that do not contain \(1_T\). We show that if the depth of \(T\) is greater or equal to \(2n + [ n \log n ]\) then \(B_T^n > A_T^n\).
- Research article
- Full Text
- Ars Combinatoria
- Volume 091
- Pages: 87-95
- Published: 30/04/2009
The boundedness and compactness of the weighted composition operator from logarithmic Bloch spaces to a class of weighted-type spaces are studied in this paper.
- Research article
- Full Text
- Ars Combinatoria
- Volume 091
- Pages: 65-82
- Published: 30/04/2009
S.M. Lee proposed the conjecture: for any \(n > 1\) and any permutation \(f\) in \(S(n)\), the permutation graph \(P(P_n, f)\) is graceful. For any integer \(n > 1\), we discuss gracefulness of the permutation graphs \(P(P_n, f)\) when \(f = (123), (n-2, n-1, n), (i, i+1), 1 \leq i \leq n-1, (12)(34)\ldots(2m-1, 2m), 1 \leq m \leq \frac{n}{2}\), and give some general results.
- Research article
- Full Text
- Ars Combinatoria
- Volume 091
- Pages: 53-64
- Published: 30/04/2009
A double-loop network (DLN) \(G(N;r,s)\) is a digraph with the vertex set \(V = \{0,1,\ldots, N-1\}\) and the edge set \(E=\{v \to v+r \pmod{N} \text{ and } v \to v+s \pmod{N} | v \in V\}\). Let \(D(N;r,s)\) be the diameter of \(G(N;r,s)\) and let us define \(D(N) = \min\{D(N;r,s) | 1 \leq r < s < N \text{ and } \gcd(N,r,s) = 1\}\), \(D_1(N) = \min\{D(N;1,s) | 1 < s 0\)). Coppersmith proved that there exists an infinite family of \(N\) for which the minimum diameter \(D(N) \geq \sqrt{3N} + c(\log N)^{\frac{1}{4}}\), where \(c\) is a constant.
- Research article
- Full Text
- Ars Combinatoria
- Volume 091
- Pages: 33-51
- Published: 30/04/2009
In this paper, we consider cycle-partition problems which deal with the case when both vertices and edges are specified and we require that they should belong to different cycles. Minimum degree and degree sum conditions are given, which are best possible.
- Research article
- Full Text
- Ars Combinatoria
- Volume 091
- Pages: 11-18
- Published: 30/04/2009
In this paper, we consider the relationships between the second order linear recurrences and the permanents and determinants of tridiagonal matrices.
- Research article
- Full Text
- Ars Combinatoria
- Volume 091
- Pages: 3-9
- Published: 30/04/2009
We correct and improve results from a recent paper by G. Ren and U. Kahler, which characterizes the Bloch, the little Bloch and Besov space of harmonic functions on the unit ball \({B} \subset \mathbb{R}^n\).
- Research article
- Full Text
- Journal of Combinatorial Mathematics and Combinatorial Computing
- Volume 069
- Pages: 125-138
- Published: 31/05/2009
Many approaches to drawing graphs in the plane can be formulated and solved as mathematical programming problems. Here, we consider only drawings of a graph where each edge is drawn as a straight-line segment, and we wish to minimize the number of edge crossings over all such drawings. Some formulations of this problem are presented that lead very naturally to other unsolved problems, some solutions, and some new open problems associated with drawing nonplanar graphs in the plane.
- Research article
- Full Text
- Journal of Combinatorial Mathematics and Combinatorial Computing
- Volume 092
- Pages: 93-101
- Published: 28/02/2015
In this paper we introduce right angle path and layer of an array. We construct Kolakoski array and study some combinatorial proper-ties of Kolakoski array. Also we obtain recurrence relation for layers and special elements.
- Research article
- Full Text
- Journal of Combinatorial Mathematics and Combinatorial Computing
- Volume 092
- Pages: 25-38
- Published: 28/02/2015
An \({eternal \;1-secure}\) set of a graph \(G = (V, E)\) is defined as a set \(S_0 \subseteq V\) that can defend against any sequence of single-vertex attacks by means of single guard shifts along edges of \(G\). That is, for any \(k\) and any sequence \(v_1, v_2, \ldots, v_k\) of vertices, there exists a sequence of guards \(u_1, u_2, \ldots, u_k\) with \(u_i \in S_{i-1}\) and either \(u_i = v_i\) or \(u_iv_i \in E\), such that each set \(S_i = (S_{i-1} -\{u_i\}) \cup \{v_i\}\) is dominating. It follows that each \(S_i\) can be chosen to be an eternal 1-secure set. The \({eternal \;1-security\; number}\), denoted by \(\sigma_1(G)\), is defined as the minimum cardinality of an eternal 1-secure set. This parameter was introduced by Burger et al. [3] using the notation \(\gamma_\infty\). The \({eternal \;m-security}\) number \(\sigma_m(G)\) is defined as the minimum number of guards to handle an arbitrary sequence of single attacks using multiple-guard shifts. A suitable placement of the guards is called an \({eternal\; m-secure}\) set. It was observed that \(\gamma(G) \leq \sigma_m(G) \leq \beta(G)\). In this paper, we obtain specific values of \(\sigma_m(G)\) for certain classes of graphs, namely circulant graphs, generalized Petersen graphs, binary trees, and caterpillars.
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