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.

Wanzhou Ye1
1Department of Mathematics, Shanghai University, Shanghai 200444
Abstract:

In this paper, we obtain a fundamental result on the dynamical behavior of symmetric weighted mappings for two-dimensional real sequence spaces \({R}_s\).

Xue-gang Chen1, Moo Young Sohn2
1Mathematics, North China Electric Power University Beijing 102206, China
2Mathematics, Changwon National University Changwon 641-773, Korea
Abstract:

In \(2006\), Mojdeh and Jafari Rad [On the total domination critical graphs, Electronic Notes in Discrete Mathematics, 24 (2006), 89-92] gave an open problem: Does there exist a \(3\)-\(\gamma_t\)-critical graph \(G\) of order \(\Delta(G) + 3\) with \(\Delta(G)\) odd and \(\delta(G) \geq 2\)? In this paper, we positively answer that for each odd integer \(n \geq 9\), there exists a \(3\)-\(\gamma_t\)-critical graph \(G_n\) of order \(n+3\) with \(\delta(G) \geq 2\). On the contrary, we also prove that for \(\Delta(G) = 3, 5, 7\), there is no \(3\)-\(\gamma_t\)-critical graph of order \(\Delta(G) + 3\) with \(\delta(G) \geq 2\).

Hong Hu1
1Department of Mathematics, Huaiyin Teachers College, Huaian 223300, Jiangsu Province, P.R.China
Abstract:

Let \(\{w_n\}\) be a second-order recurrence sequence. According to the definition and characteristics of the recurrent sequence, we proved a recursion formula for certain reciprocal sums whose denominators are products of consecutive elements of \(\{w_n\}\).

Shanhai Li1,2, Hao Shen1
1Department of Mathematics, Shanghai JiaoTong University Shanghai 200240 China
2School of Statistics and Mathematics, Shandong Economic University, Jinan Shandong 250014 China
Abstract:

Let \(G\) be a graph in which each vertex has been colored using one of \(k\) colors, say \(c_1, c_2, \ldots, c_k\). If an \(m\)-cycle \(C\) in \(G\) has \(n_i\) vertices colored \(c_i\), \(i = 1, 2, \ldots, k\), and \(|n_i – n_j| \leq 1\) for any \(i, j \in \{1, 2, \ldots, k\}\), then \(C\) is equitably \(k\)-colored. An \(m\)-cycle decomposition \(\mathcal{C}\) of a graph \(G\) is equitably \(k\)-colorable if the vertices of \(G\) can be colored so that every \(m\)-cycle in \(\mathcal{C}\) is equitably \(k\)-colored. For \(m = 4, 5\), and \(6\), we completely settle the existence problem for equitably \(2\)-colorable \(m\)-cycle decompositions of complete graphs with the edges of a \(1\)-factor added.

AP Burger1, JH Van Vuuren1
1Department of Logistics, University of Stellenbosch, Private Bag X1, Matieland, 7602, Republic of South Africa,
Abstract:

Suppose a network facility location problem is modelled by means of an undirected, simple graph \(G = (\mathcal{V, E})\) with \(\mathcal = \{v_1, \ldots, v_n\}\). Let \(\mathbf{r} = (r_1, \ldots, r_n)\) and \(\mathbf{s} = (s_1, \ldots, s_n)\) be vectors of nonnegative integers and consider the combinatorial optimization problem of locating the minimum number, \(\gamma(\mathbf{r}, \mathbf{s}, G)\) (say), of commodities on the vertices of \(G\) such that at least \(s_j\) commodities are located in the vicinity of (i.e. in the closed neighbourhood of) vertex \(v_j\), with no more than \(r_j\) commodities placed at vertex \(v_j\) itself, for all \(j = 1, \ldots, n\). In this paper we establish lower and upper bounds on the parameter \(\gamma(\mathbf{r}, \mathbf{s}, G)\) for a general graph \(G\). We also determine this parameter exactly for certain classes of graphs, such as paths, cycles, complete graphs, complete bipartite graphs and establish good upper bounds on \(\gamma(\mathbf{r}, \mathbf{s}, G)\) for a class of grid graphs in the special case where \(r_j = r\) and \(s_j = s\) for all \(j = 1, \ldots, n\).

Gregory P.Tollisen1, Tamas Lengyel2
1OCCIDENTAL COLLEGE, MATHEMATICS DEPARTMENT, 1600 CamPus Roab, Los ANGELES, CA 90041
2OCCIDENTAL COLLEGE, MaTHEMATICS DEPARTMENT, 1600 Campus Roab, Los ANGELES, CA 90041
Abstract:

Let \(A\) be an arbitrary circulant stochastic matrix, and let \(\underline{x}_0\) be a vector. An “asymptotic” canonical form is derived for \(A^k\) (as \(k \to \infty\)) as a tensor product of three simple matrices by employing a pseudo-invariant on sections of states for a Markov process with transition matrix \(A\), and by analyzing how \(A\) acts on the sections, through its auxiliary polynomial. An element-wise asymptotic characterization of \(A^k\) is also given, generalizing previous results to cover both periodic and aperiodic cases. For a particular circulant stochastic matrix, identifying the intermediate stage at which fractions first appear in the sequence \(\underline{x}_k = A^k \underline{x}_0\), is accomplished by utilizing congruential matrix identities and \((0,1)\)-matrices to determine the minimum \(2\)-adic order of the coordinates of \(\underline{x}_k\), through their binary expansions. Throughout, results are interpreted in the context of an arbitrary weighted average repeatedly applied simultaneously to each term of a finite sequence when read cyclically.

Yan Jin1, Zhao Kewen2,3, Hong-Jian Lai4, Ju Zhou4
1School of Mathematics and Systems Sciences, Shandong University, Jinan 250100, P. R. China
2Department of Mathematics, Qiongzhou University, Wuzhishan, Hainan 572200, P. R. China
3Department of Mathematics, Hainan Normal University, Haikou, Hainan 571100, P. R. China
4Department of Mathematics, West Virginia University, Morgantown, WV 26506- 6310, USA
Abstract:

A graph \(G\) is \(s\)-Hamiltonian if for any \(S \subseteq V(G)\) of order at most \(s\), \(G-S\) has a Hamiltonian cycle, and \(s\)-Hamiltonian connected if for any \(S \subseteq V(G)\) of order at most \(s\), \(G-S\) is Hamiltonian-connected. Let \(k > 0, s \geq 0\) be two integers. The following are proved in this paper:(1) Let \(k \geq s+2\) and \(s \leq n-3\). If \(G\) is a \(k\)-connected graph of order \(n\) and if \(\max\{d(v) : v \in I\} \geq (n+s)/2\) for every independent set \(I\) of order \(k-s\) such that \(I\) has two distinct vertices \(x,y\) with \(1 \leq |N(x) \cap N(y)| \leq \alpha(G)+s-1\), then \(G\) is \(s\)-Hamiltonian.(2) Let \(k \geq s+3\) and \(s \leq n-2\). If \(G\) is a \(k\)-connected graph of order \(n\) and if \(\max\{d(v) : v \in I\} \geq (n+s+1)/2\) for every independent set \(I\) of order \(k-s-1\) such that \(I\) has two distinct vertices \(x,y\) with \(1 \leq |N(x) \cap N(y)| \leq \alpha(G)+s\), then \(G\) is \(s\)-Hamiltonian connected.These results extend several former results by Dirac, Ore, Fan, and Chen.

Bart De Bruyn1
1Department of Pure Mathematics and Computer Algebra, Ghent University, Galglaan 2, B-9000 Gent, Belgium,
Abstract:

We show that every generalized quadrangle of order \((4,6)\) with a spread of symmetry is isomorphic to the Ahrens-Szekeres generalized quadrangle \(\text{AS}(5)\). It then easily follows that every generalized quadrangle of order \(5\) with an axis of symmetry is isomorphic to the classical generalized quadrangle \(\text{Q}(4, 5)\).

A. Iranmanesh1, Y. Pakravesh1, A. Mahmiani2
1Department of Mathematice, Tarbiat Modares Universiry P. O. Box: 14115-137, Tehran, Iran
2University of Payame Noor, Gonbade Kavoos, Iran
Abstract:

A \(C_5C_7\) net is a trivalent decoration made by alternating pentagons \(C_5\) and heptagons \(C_7\). It can cover either a cylinder or a torus. In this paper, we compute the Szeged index of \(HC_5C_7[ r, p ]\) nanotube.

M. Abreu1, M. Funk1, D. Labbate2, V. Napolitano3
1 Dipartimento di Matematica, Universita della Basilicata, Viale dell’ Ateneo Lucano, 85100 Potenza, Italy.
2Dipartimento di Matematica, Politecnico di Bari, Via E, Orabona, 4, 70125 Bari, Italy.
3Dipartimento di Matematica, Universita della Basilicata, Viale dell’ Ateneo Lucano, 85100 Potenza, Italy.
Abstract:

We present algebraic constructions yielding incidence matrices for all finite Desarguesian elliptic semiplanes of types \(C, D\), and \(L\). Both basic ingredients and suitable notations are derived from addition and multiplication tables of finite fields. This approach applies also to the only elliptic semiplane of type B known so far. In particular, the constructions provide intrinsic tactical decompositions and partitions for these elliptic semiplanes into elliptic semiplanes of smaller order.

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