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.

Changqing Xu1, Yatao Du2
1Department of Applied Mathematics, Hebei University of Technology Tianjin, 300130, China
2 Department of Mathematics, Shijiazhuang Mechanical Engineering College Shijiazhuang 050003, China
Abstract:

A \((g, f)\)-coloring is a generalized edge-coloring in which each color appears at each vertex \(v\) at least \(g(v)\) and at most \(f(v)\) times, where \(g(v)\) and \(f(v)\) are nonnegative and positive integers assigned to each vertex \(v\), respectively. The minimum number of colors used by a \((g, f)\)-coloring of \(G\) is called the \((g, f)\)-chromatic index of \(G\). The maximum number of colors used by a \((g, f)\)-coloring of \(G\) is called the upper \((g, f)\)-chromatic index of \(G\). In this paper, we determine the \((g, f)\)-chromatic index and the upper \((g, f)\)-chromatic index in some cases.

B. Manoochehrian1, H. Yousefi-Azari2, A. R. Ashrafi3
1Academic Center for Education, Culture and Research, Tehran Branch, Tehran, 1. R. Iran
2Center of Excellence in Biomathematics, School of Mathematics, Statistics and Computer Science, University of Tehran, Tehran, I. R. Iran
3Department of Mathematics, Faculty of Science, University of Kashan, Kashan 87317-51167, 1. R. Iran
Abstract:

The Szeged index extends the Wiener index for cyclic graphs by counting the number of atoms on both sides of each bond and summing these counts. This index was introduced by Ivan Gutman at the Attila Jozsef University in Szeged in \(1994\), and is thus called the Szeged index. In this paper, we introduce a novel method for enumerating by cuts. Using this method, an exact formula for the Szeged index of a zig-zag polyhex nanotube \(T = TUHC_6{[p,q]}\) is computed for the first time.

Pinar Anapa1, ibrahim Gunaltili1
1Eskisehir Osmangazi University Departmant of Mathematics 26480 Eskisehir-Tiirkiye
Abstract:

In this study, we showed that an \((n+1)\)-regular linear space, which is the complement of a linear space having points not on \(m+1\) lines such that no three are concurrent in a projective subplane of odd order \(m\), \(m \geq 9\), could be embedded into a projective plane of order \(n\) as the complement of Ostrom’s hyperbolic plane.

H. Fujii1, M. Sawa1
1Graduate School of Information Science, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, 464-8601, Japan.
Abstract:

For general graphs \(G\), it is known \([6]\) that the minimal length of an addressing scheme, denoted by \(N(G)\), is less than or equal to \(|G| – 1\). In this paper, we prove that for almost all complete bipartite graphs \(K_{m,n}\), \(N(K_{m,n}) = |K_{m,n}| – 2\).

Zongtian Wei1, Shenggeui Zhang2
1Department of Mathematics, Xi’an University of Architecture and Technology, Xi’an, Shaanxi 710055, P.R. China
2Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an, Shaanxi 710072, P.R. China
Abstract:

A vertex subversion strategy of a graph \(G\) is a set of vertices \(X \subseteq V(G)\) whose closed neighborhood is deleted from \(G\). The survival subgraph is denoted by \(G/X\). The vertex-neighbor-integrity of \(G\) is defined to be \(VNI(G) = \min\{|X| + r(G/X) : X \subseteq V(G)\},\) where \(r(G/X)\) is the order of a largest component in \(G/X\). This graph parameter was introduced by Cozzens and Wu to measure the vulnerability of spy networks. It was proved by Gambrell that the decision problem of computing the vertex-neighbor-integrity of a graph is NP-complete. In this paper, we evaluate the vertex-neighbor-integrity of the composition graph of two paths.

M.M. Shikare1, B.N. Waphare 1
1Department of Mathematics University of Pune, Pune – 411007 (India)
Abstract:

In this paper, we prove that a matroid with at least two elements is connected if and only if it can be obtained from a loop by a nonempty sequence of non-trivial single-element extensions and series extensions.

Pierre Ille1, William Kocay2
1Institut de Mathémathiques de Luminy CNRS — UMR 6206 163 avenue de Luminy, Case 907 13288 Marseille Cedex 9, France
2Computer Science Department St. Paul’s College, University of Manitoba Winninpeg, MB, Canada R3T 2N2
Abstract:

Let \(G\) and \(H\) be graphs with a common vertex set \(V\), such that \(G – i \cong H – i\)for all \(i \in V\). Let \(p_i\) be the permutation of \(V – i\) that maps \(G – i\) to \(H – i\), and let \(q_i\) denote the permutation obtained from \(p_i\) by mapping \(i\) to \(i\). It is shown that certain algebraic relations involving the edges of \(G\) and the permutations \(q_iq_j^{-1}\) and \(q_iq_k^{-1}\), where \(i, j, k \in V\) are distinct vertices, often force \(G\) and \(H\) to be isomorphic.

Tan Mingshu1
1Department of Mathematics, Chongqing Three Gorges University, Chongqing Wanzhou 404000, People’s Republic of China
Abstract:

The factorization of matrix \(A\) with entries \(a_{i,j}\) determined by \(a_{i,j} = \alpha a_{i-1,j-1} + \beta a_{i,j-1}\) is derived as \(A = TP^T\). An interesting factorization of matrix \(B\) with entries \(b_{i,j} = \alpha b_{i-1,j} + \beta b_{i,j-1}\) is given by \(B = P[\alpha]TP^{T}[\beta]\). The beautiful factorization of matrix \(C\) whose entries satisfy \(c_{i,j} = \alpha c_{i-1,j} + \beta c_{i-1,j-1} + Ye_{i-1,j-1}\) is founded to be \(C = P[\alpha]DP^T[\beta]\), where \(T\) is a Toeplitz matrix, and \(P\) and \(P[\alpha]\) are Pascal matrices. The matrix product factorization to the problem is solved perfectly so far.

D. Bauer1, N. Kahl2, L. Mcguire3, E. Schmeichel4
1Department of Mathematical Sciences Stevens Institute of Technology Hoboken, NJ 07030
2 Department of Mathematics and Computer Science Seton Hall University South Orange, NJ 07079
3Department of Mathematical Sciences Muhlenberg College Allentown, PA 18104
4 Department of Mathematics San Jose State University San Jose, CA 95192
Abstract:

Dirac showed that a \(2\)-connected graph of order \(n\) with minimum degree \(\delta\) has circumference at least \(\min\{2\delta, n\}\). We prove that a \(2\)-connected, triangle-free graph \(G\) of order \(n\) with minimum degree \(\delta\) either has circumference at least \(\min\{4\delta – 4, n\}\), or every longest cycle in \(G\) is dominating. This result is best possible in the sense that there exist bipartite graphs with minimum degree \(\delta\) whose longest cycles have length \(4\delta – 4\), and are not dominating.

Jian-Liang Wu 1, Yu-Liang Wu2
1School of Mathematics, Shandong University, Jinan, 250100, China
2Department of Computer Science and Engineering The Chinese University of Hong Kong, Hong Kong
Abstract:

The vertex linear arboricity \(vla(G)\) of a graph \(G\) is the minimum number of subsets into which the vertex set \(V(G)\) can be partitioned so that each subset induces a subgraph whose connected components are paths. It is proved here that \(\lceil \frac{\omega(G)}{2}\rceil \leq vla(G) \leq \lceil \frac{\omega(G)+1}{2}\rceil\) for a claw-free connected graph \(G\) having \(\Delta(G) \leq 6\), where \(\omega(G)\) is the clique number of \(G\).

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