Journal of Combinatorial Mathematics and Combinatorial Computing

ISSN: 0835-3026 (print) 2817-576X (online)

The Journal of Combinatorial Mathematics and Combinatorial Computing (JCMCC) embarked on its publishing journey in April 1987. From 2024 onward, it publishes four volumes per year in March, June, September and December. JCMCC has gained recognition and visibility in the academic community and is indexed in renowned databases such as MathSciNet, Zentralblatt, Engineering Village and Scopus. The scope of the journal includes; Combinatorial Mathematics, Combinatorial Computing, Artificial Intelligence and applications of Artificial Intelligence in various files.

K.M. Martin1, Jennifer Seberry2, BR. Wild1
1Department of Mathematics RHBNC Egham Hill, Egham Surrey TW20 0EX
2Department of Computer Science University of Wollongong NSW, 2500, Australia
Abstract:

We consider certain resolvable designs which have applications to doubly perfect Cartesian authentication schemes. These generalize structures determined by sets of mutually orthogonal Latin squares and are related to semi-Latin squares and other designs which find applications in the design of experiments.

Alan Rahilly1
1Department of Mathematics University of Queensland, St. Lucia 4067 Australia
Abstract:

A \(1\)-spread of a BIBD \(\mathcal{D}\) is a set of lines of maximal size of \(\mathcal{D}\) which partitions the point set of \(\mathcal{D}\). The existence of infinitely many non-symmetric BIBDs which (i) possess a \(1\)-spread, and (ii) are not merely a multiple of a symmetric BIBD,
is shown. It is also shown that a \(1\)-spread \(\mathcal{S}\) gives rise to a regular group divisible design \(\mathcal{G}(\mathcal{S})\). Necessary and sufficient conditions that the dual of such a group divisible design \(\mathcal{G}(\mathcal{S})\) be a group divisible design are established and used to show the existence of an infinite class of symmetric regular group divisible designs whose duals are not group divisible.

Linda M. Lawson1, Teresa W. Haynes2
1Department of Mathematics East Tennessee State University Johnson City, TN 37614
2Department of Computer Science East Tennessee State University Johnson City, TN 37614
Abstract:

We consider the changing and unchanging of the edge covering and edge independence numbers of a graph when the graph is modified by deleting a node, deleting an edge, or adding an edge. In this paper, we present characterizations for the graphs in each of these classes and some relationships among them.

Earl S. Kramer1, Spyros S. Magliveras1, Tran van Trung1
1University of Nebraska Lincoln, Nebraska
Abstract:

Let \(G\) be the automorphism group of an \((3, 5, 26)\) design. We show the following: (i) If \(13\) divides \(|G|\), then \(G\) is a subgroup of \(Z_2 \times F_{r_{13 \cdot 12}}\), where \(F_{r_{13 \cdot 12}}\) is the Frobenius group of order \(13 \cdot 12\); (ii)If \(5\) divides \(|G|\), then \(G \cong {Z}_5\) or \(G \cong {D}_{10}\); and (iii) Otherwise, either \(|G|\) divides \(3 \cdot 2^3\) or \(2^4\).

Sin-Min Lee1, Sheng-Ping Lo2, Eric Seah3
1Dept. of Mathematics and Computer Science San Jose State University San Jose, California 95192 U.S.A.
2AT & T, Bell Laboratories Holmdel, New Jersey 07733 U.S.A,
3Dept. of Actuarial and Management Sciences University of Manitoba Winnipeg, Manitoba R3T 2N2 CANADA
Abstract:

We investigate the edge-gracefulness of \(2\)-regular graphs.

Michael A. Henning1
1University of Natal Pietermaritzburg, 3200 South Africa
Abstract:

For \(n\) a positive integer and \(v\) a vertex of a graph \(G\), the \(n\)th order degree of \(v\) in \(G\), denoted by \(\text{deg}_n(v)\), is the number of vertices at distance \(n\) from \(v\). The graph \(G\) is said to be \(n\)th order regular of degree \(k\) if, for every vertex \(v\) of \(G\), \(\text{deg}_n(v) = k\). For \(n \in \{7, 8, \ldots, 11\}\), a characterization of \(n\)th order regular trees of degree \(2\) is obtained. It is shown that, for \(n \geq 2\) and \(k \in \{3, 4, 5\}\), if \(G\) is an \(n\)th order regular tree of degree \(k\), then \(G\) has diameter \(2n – 1\).

M.J. Grannell1, T.S. Griggs1, R.A. Mathon2
1Department of Mathematics and Statistics Lancashire Polytechnic Preston PRI 2TQ United Kingdom
2Department of Computer Science University of Toronto Toronto, Ontario, M5S 1A4 Canada
Abstract:

We prove that there exist precisely \(459\) pairwise non-isomorphic Steiner systems \(S(5,6,48)\) stabilized by the group \({PSL}_2(47)\).

Stanley E. Payne1
1Department of Mathematics University of Colorado at Denver Denver, CO. 80217
Abstract:

The known generalized quadrangles with parameters \((s,t)\) where \(|s-t| = 2\) have been characterized in several ways by M. De Soete \([D]\), M. De Soete and J. A. Thas \([DT1]\), \([DT2]\), \([DT4]\), and the present author \([P]\). Certain of these results are interpreted for a coset geometry construction.

Cantian Lin1, Haiping Lin2, W. D. Wallis2, J. L. Yucas2
1Department of Mathematical Sciences University of Nevada Las Vegas, NV, 89154
2Department of Mathematics Southern Illinois University Carbondale, IL, 62901
Abstract:

In this paper, we illustrate the relationship between profiles of Hadamard matrices and weight distributions of codes, give a new and efficient method to determine the minimum weight \(d\) of doubly even self-dual \([2n,n,d]\) codes constructed by using Hadamard matrices of order \(n = 8t + 4\) with \(t \geq 1\), and present a new proof that the \([2n,n,d]\) codes have \(d \geq 8\) for all types of Hadamard matrices of order \(n = 8t + 4\) with \(t \geq 1\). Finally, we discuss doubly even self-dual \([72,36,d]\) codes with \(d = 8\) or \(d = 12\) constructed by using all currently known Hadamard matrices of order \(n = 36\).

Abstract:

We define an \({extremal \; graph}\) on \(v\) vertices to be a graph that has the maximum number of edges on \(v\) vertices, and that contains neither \(3\)-cycles nor \(4\)-cycles.
We establish that every vertex of degree at least \(3\), in an extremal graph of at least \(7\) vertices, is in a \(5\)-cycle; we enumerate all of the extremal graphs on \(21\) or fewer vertices; and we determine the size of extremal graphs of orders \(25\), \(26\), and \(27\).

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