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.

S.M. Anvariyeh1, S. Mirvakili2, B. Davvaz1
1Department of Mathematics, Yazd University, Yazd, Iran
2Department of Mathematics, Payame Noor University, Yazd, Iran
Abstract:

In this paper, the class of \((m,n)\)-ary hypermodules is introduced and several properties and examples are found. \((m,n)\)-ary hypermodules are a generalization of hypermodules. On the other hand, we can consider \((m,n)\)-ary hypermodules as a good generalization of \((m,n)\)-ary modules. We define the fundamental relation \(\epsilon^*\) on the \((m,n)\)-ary hypermodules \(M\) as the smallest equivalence relation such that \(M/\epsilon^*\) is an \((m,n)\)-ary module, and then some related properties are investigated.

Zehui Shao1, Xiaodong Xu2, Qiquan Bao3
1School of Information Science & Technology, Chengdu University, Chengdu, 610106, China
2Guangxi Academy of Sciences, Nanning, Guangxi 530007,China
3Department of Control Science and Engineering, Huazhong University of Science and Technology, Wuhan 430074, China
Abstract:

For given graphs \(G_1\) and \(G_2\), the Ramsey number \(R(G_1, G_2)\) is defined to be the least positive integer \(n\) such that every graph \(G\) on \(n\) vertices, either \(G\) contains a copy of \(G_1\) or the complement of \(G\) contains a copy of \(G_2\). In this note, we show that \(R(C_m, B_n) = 2m-1\) for \(m \geq 2n-1 \geq 7\). With the help of computers, we obtain the exact values of \(14\) small cycle-book Ramsey numbers.

Song Guo1
1School of Mathematical Science, Huaiyin Normal University Huaian 223300, People’s Republic of China
Abstract:

For positive integers \(c \geq 0\) and \(k \geq 1\), let \(n = R(c, k)\) be the least integer, provided it exists, such that every \(2\)-coloring of the set \([1,n] = \{1,\ldots,n\}\) admits a monochromatic solution to the equation \(x + y+c = 4z\) with \(x, y, z \in [1,n]\). In this paper, the precise value of \(R(c, 4)\) is shown to be \(\left\lceil{3c + 2}/{8}\right\rceil\) for all even \(c \geq 34\).

Nihal Yilmaz Ozgur1
1BatikeEsin UNIVERSITY, DEPARTMENT OF MATHEMATICS, 10145 BALIKESIR, TURKEY
Abstract:

Given a positive integer \(n\) such that \(-1\) is a quadratic residue mod \(n\), we give an algorithm that computes the integers \(u\) and \(v\) which satisfy the equation \(n = u^2 + v^2\). To do this, we use the group structure of the Modular group \(\Gamma= \text{PSL}(2,\mathbb{Z})\).

Min-Jen Jou1
1Department of Insurance Ling Tung University Taichung, Taiwan 40852, R.O.C.
Abstract:

For a graph \(G = (V(G),E(G))\), a set \(S \subseteq V(G)\) is called a dominating set if \(N_G[S] = V(G)\). A dominating set \(S\) is said to be minimal if no proper subset \(S’ \subset S\) is a dominating set. Let \(\gamma(G)\) (called the domination number) and \(\Gamma(G)\) (called the upper domination number) be the minimum cardinality and the maximum cardinality of a minimal dominating set of \(G\), respectively. For a tree \(T\) of order \(n \geq 2\), it is obvious that \(1 = \gamma(K_{1,n-1}) \leq \gamma(T) \leq \Gamma(T) \leq \Gamma(K_{1,n-1}) = n-1\). Let \(t(n) = \min_{|T|=n}(\Gamma(T)-\gamma(T))\). In this paper, we determine \(t(n)\) for all natural numbers \(n\). We also characterize trees \(T\) with \(\Gamma(T) – \gamma(T) = t(n)\).

Shi-Mei Ma1
1Department of Information and Computing Science, Northeastern University at Qinhuangdao, Hebei 066004, China
Abstract:

The signless \(r\)-associated Stirling numbers of the first kind \(d_r(n, k)\) counts the number of permutations of the set \(\{1,2,\ldots,n\}\) that have exactly \(k\) cycles, each of which is of length greater than or equal to \(r\), where \(r\)is a fixed positive integer. F. Brenti obtained that the generating polynomials of the numbers \(d_r(n, k)\) have only real zeros. Here we consider the location of zeros of these polynomials.

Chin-Mei Fu1, Wen-Chung Huang2
1Department of Mathematics Tamkang University, Tamsui, Taipei Shien, Taiwan, Republic of China
2Department of Mathematics Soochow University Taipei, Taiwan, Republic of China
Abstract:

A kite-design of order \(n\) is a decomposition of the complete graph \(K_n\) into kites. Such systems exist precisely when \(n \equiv 0,1 \pmod{8}\). Two kite systems \((X,\mathcal{K}_1)\) and \((X,\mathcal{K}_2)\) are said to intersect in \(m\) pairwise disjoint blocks if \(|\mathcal{K}_1 \cap \mathcal{K}_2| = m\) and all blocks in \(\mathcal{K}_1 \cap \mathcal{K}_2\) are pairwise disjoint. In this paper, we determine all the possible values of \(m\) such that there are two kite-designs of order \(n\) intersecting in \(m\) pairwise disjoint blocks, for all \(n \equiv 0,1 \pmod{8}\).

Lihua Feng1, Guihai Yu1
1School of Mathematics, Shandong Institute of Business and Technology 191 Binhaizhong Road, Yantai, Shandong, P.R. China, 264005.
Abstract:

In this note, we present some upper bounds for the \(k\)th largest eigenvalue of the adjacency matrix as well as the Laplacian matrix of graphs. Special attention is paid to the Laplacian matrix of trees.

A.M. Khalaf1, Y.H. Peng1
1Department of Mathematics, Faculty of Science Universiti Putra Malaysia, 43400 Serdang, Selangor, Malaysia
Abstract:

Let \(P(G, \lambda)\) denote the chromatic polynomial of a graph \(G\). Two graphs \(G\) and \(H\) are chromatically equivalent, written \(G \sim H\), if \(P(G, \lambda) = P(H, \lambda)\). A graph \(G\) is chromatically unique, written \(x\)-unique, if for any graph \(H\), \(G \sim H\) implies that \(G\) is isomorphic with \(H\). In this paper, we prove that the graph \(\theta(a_1, a_2, \ldots, a_6)\) is \(x\)-unique for exactly two distinct values of \(a_1, a_2, \ldots, a_6\).

Liangxia Wan1, Yanpei Liu1
1Department of Mathematics Beijing Jiaotong University, Beijing 100044, P.R.China
Abstract:

In this paper, we give an explicit expression of the genus distributions of \(M_j^n\), for \(j = 1, 2, \ldots, 11\), which are introduced in the previous paper “Orientable embedding distributions by genus for certain types of non-planar graphs”. For a connected graph \(G = (V, E)\) with a cycle, let \(e\) be an edge on a cycle. By adding \(2n\) vertices \(u_1, u_2,u_3 \ldots, u_n, v_1, v_2,v_3 \ldots, v_n\) on \(e\) in sequence and connecting \(u_k, v_k\) for \(1 \leq k \leq n\), a non-planar graph \(G_n\) is obtained for \(n \geq 3\). Thus, the orientable embedding distribution of \(G_n\) by genus is obtained via the genus distributions of \(M_j^n\).

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