Ars Combinatoria

ISSN 0381-7032 (print), 2817-5204 (online)

Ars Combinatoria is the oldest Canadian Journal of Combinatorics, established in 1976. The journal is dedicated to advancing the field of combinatorial mathematics through the publication of high-quality research papers. From 2024 onward, it publishes four volumes per year in March, June, September and December. Ars Combinatoria has gained recognition and visibility in the academic community and is indexed in renowned databases such as MathSciNet, Zentralblatt, and Scopus. The Scope of the journal includes Graph theory, Design theory, Extremal combinatorics, Enumeration, Algebraic combinatorics, Combinatorial optimization, Ramsey theory, Automorphism groups, Coding theory, Finite geometries, Chemical graph theory but not limited.

Ahmed Ainouche1
1CEREGMIA-GRIMAAG UAG-Campus de Schoelcher B.P. 7209 97275 Schoelcher Cedex Martinique (FRANCE)
Abstract:

A simple, undirected \(2\)-connected graph \(G\) of order \(n\) belongs to the class \(\mathcal{B}(n,\theta)\), \(\theta \geq 0\) if \(2(d(x) + d(y) + d(z)) \geq 3(n – 1 – \theta)\) holds for all independent triples \(\{x,y,z\}\) of vertices. It is known (Bondy’s theorem for \(2\)-connected graphs) that \(G\) is hamiltonian if \(\theta = 0\). In this paper we give a full characterization of graphs \(G\) in \(\mathcal{B}(n,\theta)\), \(\theta \leq 2\) in terms of their dual hamiltonian closure.

John Martino1, Paula Smith2
1Western Michigan University
2Ohio Dominican University
Abstract:

Two classes of regular Cayley maps, balanced and antibalanced, have long been understood, see \([12,11]\). A recent generalization is that of an e-balanced map, see \([7,2,5,8]\). These maps can be described using the power function introduced in \([4]\); e-balanced maps are the ones with constant power functions on the generating set. In this paper we examine a further generalization to the situation where the power function alternates between two values.

E.Gokcen Kocer1, Toufik Mansour2, Naim Tuglu3
1Faculty of Education, University of Selcuk, 42099 Meram-Konya, Turkey
2Department of Mathematics, University of Haifa, 31905 Haifa, Israel
3Department of Mathematics, University of Gazi, 06500 Teknikokullar-Ankara, Turkey
Abstract:

In this paper, we obtain the spectral norm and eigenvalues of circulant matrices with Horadam’s numbers. Furthermore, we define the semicirculant matrix with these numbers and give the Euclidean norm of this matrix.

V Vijayalakshmi1
1Department of Mathematics Anna University MIT Campus, Chennai – 600 044, India
Abstract:

We denote by \(G(n)\) the graph obtained by removing a Hamilton cycle from the complete graph \(K_n\). In this paper, we calculate the lower bound for the minimum number of monochromatic triangles in any \(2\)-edge coloring of \(G(n)\) using the weight method. Also, by explicit constructions, we give an upper bound for the minimum number of monochromatic triangles in \(2\)-edge coloring of \(G(n)\) and the difference between our lower and upper bound is just two.

Akhlaq Ahmad Bhatti1
1SCHOOL OF MATHEMATICAL SCIENCES 35-C-II, GULBERG III, LAHORE, PAKISTAN
Abstract:

In this paper, it is proved that the \(h\)-chromatic uniqueness of the linear \(h\)-hypergraph consisting of two cycles of lengths \(p\) and \(q\) having \(r\) edges in common when \(p=q\), \(2 \leq r \leq p-2\), and \(h \geq 3\). We also obtain the chromatic polynomial of a connected unicyclic linear \(h\)-hypergraph and show that every \(h\)-uniform cycle of length three is not chromatically unique for \(h \geq 3\).

M. Esmaeili1, T.A. Gulliver2
1Department of Mathematical Sciences Isfahan University of Technology Isfahan, Iran
2Dept. of Electrical and Computer Engineering University of Victoria P.O. Box 3055, STN CSC Victoria, B.C., V8W 3P6 Canada
Abstract:

The projection of binary linear block codes of length \(4m\) on \(\mathbb{F}_4^m\) is considered. Three types of projections, namely projections \(SE\), \(E\), and \(O\) are introduced. The BCH codes, Golay codes, Reed-Muller codes, and the quadratic residue code \(q_{32}\) are examined.

Mehdi Eliasi1, Bijan Taeri1
1Department of Mathematical Sciences, Isfahan University of Technology, Isfehan, Iran
Abstract:

The hyper Wiener index of a connected graph \(G\) is defined as
\(WW(G) = \frac{1}{2}\sum_{u,v \in V(G)} d(u,v) + \frac{1}{2}\sum_{(u,v) \in V(G)} d(u,v)^2\) where \(d(u, v)\) is the distance between vertices \(u,v \in V(G)\).
In this paper we find an exact expression for hyper Wiener index of \(HC_6[p, q]\), the zigzag polyhex nanotori.

T.Aaron Gulliver1, John N.C.Wong1
1Department of Electrical and Computer Engineering, University of Victoria, P.O. Box 3055, STN CSC, Victoria, BC, Canada V8W 3P6,
Abstract:

In this paper, we classify all optimal linear \([n, n/2]\) codes over \(\mathbb{Z}_4\) up to length \(n = 8\), and determine the number of optimal codes which are self-dual and formally self-dual. Optimal codes with linear binary images are identified. In particular, we show that for length \(8\), there are nine optimal codes for the Hamming distance, one optimal code for the Lee distance, and two optimal codes for the Euclidean distance.

Zan-Bo Zhang1,2, Tao Wang3, Dingjun Lou1
1Department of Computer Science, Sun Yat-sen University, Guangzhou 510275, China
2Department of Computer Engineering, Guangdong Industry Technical College, Guangzhou 510300, China
3Center for Combinatorics, LPMC, Nankai University, Tianjin 300071, China
Abstract:

In this paper, we show that if \(k \geq \frac{v+2}{4}\), where \(v\) denotes the order of a graph, a non-bipartite graph \(G\) is \(k\)-extendable if and only if it is \(2k\)-factor-critical. If \(k \geq \frac{v-3}{4}\), a graph \(G\) is \(k\)-extendable if and only if it is \((2k+1)\)-factor-critical. We also give examples to show that the two bounds are best possible. Our results are answers to a problem posted by Favaron \([3]\) and Yu \([11]\).

Zongtian Wei1, Yang Li1, Junmin Zhang1
1College of Science, Xi’an University of Architecture and Technology Xian, Shaanxi 710055, P.R. China
Abstract:

The edge-neighbor-scattering number of a graph \(G\) is defined to be \(EN_S(G) = \max\limits_{S\subseteq E(G)}\{w(G/S) -\mid |S|\}\) where \(S\) is any edge-cut-strategy of \(G\), \(w(G/S)\) is the number of the components of \(G/S\). In this paper, we give edge-neighbor-scattering number of some special classes of graphs, and then mainly discuss the general properties of the parameter.

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