
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.
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- Research article
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- Ars Combinatoria
- Volume 088
- Pages: 33-45
- Published: 31/07/2008
In this paper, we give some relations involving the usual Fibonacci and generalized order-\(k\) Pell numbers. These relations show that the generalized order-\(k\) Pell numbers can be expressed as the summation of the usual Fibonacci numbers. We find families of Hessenberg matrices such that the permanents of these matrices are the usual Fibonacci numbers, \(F_{2i-1}\), and their sums. Also, extending these matrix representations, we find families of super-diagonal matrices such that the permanents of these matrices are the generalized order-\(k\) Pell numbers and their sums.
- Research article
- Full Text
- Ars Combinatoria
- Volume 088
- Pages: 27-32
- Published: 31/07/2008
Let \(G\) be a finite group and \(S\) be a subset (possibly containing the identity element) of \(G\). We define the Bi-Cayley graph \(X = BC(G, S)\) to be the bipartite graph with vertices \(G \times \{0, 1\}\) and edges \(\{(g, 0), (sg, 1) : g \in G, s \in S\}\). In this paper, we show that if \(X = BC(G, S)\) is connected, then \(\kappa(X) = \delta(X)\).
- Research article
- Full Text
- Ars Combinatoria
- Volume 088
- Pages: 21-25
- Published: 31/07/2008
Some new characterizations for harmonic Bergman space on the unit ball \({B}\) in \(\mathbb{R}^n\) are given in this paper. They can be described as derivative-free characterizations.
- Research article
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- Ars Combinatoria
- Volume 088
- Pages: 3-20
- Published: 31/07/2008
The planar Ramsey number \(PR(H_1, H_2)\) is the smallest integer \(n\) such that any planar graph on \(n\) vertices contains a copy of \(H_1\) or its complement contains a copy of \(H_2\). It is known that the Ramsey number \(R(K_4 – e, K_k – e)\) for \(k \leq 6\). In this paper, we prove that \(PR(K_4 – e, K_6 – e) = 16\) and show the lower bounds on \(PR(K_4 – e, K_k – e)\).
- Research article
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- Ars Combinatoria
- Volume 087
- Pages: 403-413
- Published: 30/04/2008
Let \(K_v\) be a complete graph with \(v\) vertices, and \(G = (V(G), E(G))\) be a finite simple graph. A \(G\)-design \(G-GD_\lambda(v)\) is a pair \((X, \mathcal{B})\), where \(X\) is the vertex set of \(K_v\), and \(\mathcal{B}\) is a collection of subgraphs of \(K_v\), called blocks, such that each block is isomorphic to \(G\) and any two distinct vertices in \(K_v\) are joined in exactly \(\lambda\) blocks of \(\mathcal{B}\). In this paper, the existence of graph designs \(G-GD_\lambda(v)\), \(\lambda > 1\), for eight graphs \(G\) with six vertices and eight edges is completely solved.
- Research article
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- Ars Combinatoria
- Volume 087
- Pages: 393-402
- Published: 30/04/2008
A \({weighted \;graph}\) is one in which every edge \(e\) is assigned a nonnegative number \(w(e)\), called the \({weight}\) of \(e\). The \({weight\; of \;a \;cycle}\) is defined as the sum of the weights of its edges. The \({weighted \;degree}\) of a vertex is the sum of the weights of the edges incident with it. In this paper, motivated by a recent result of Fujisawa, we prove that a \(2\)-connected weighted graph \(G\) contains either a Hamilton cycle or a cycle of weight at least \(2m/3\) if it satisfies the following conditions:
\((1)\) The weighted degree sum of every three pairwise nonadjacent vertices is at least \(m\);\((2)\)In each induced claw and each induced modified claw of \(G\), all edges have the same weight.This extends a theorem of Zhang, Broersma and Li.
- Research article
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- Ars Combinatoria
- Volume 087
- Pages: 385-392
- Published: 30/04/2008
The \({restricted edge-connectivity}\) of a graph is an important parameter to measure fault-tolerance of interconnection networks. This paper determines that the restricted edge-connectivity of the de Bruijn digraph \(B(d,n)\) is equal to \(2d – 2\) for \(d \geq 2\) and \(n \geq 2\) except \(B(2,2)\). As consequences, the super edge-connectedness of \(B(d,n)\) is obtained immediately.
- Research article
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- Ars Combinatoria
- Volume 087
- Pages: 377-383
- Published: 30/04/2008
An edge coloring of a graph is called \({square-free}\) if the sequence of colors on certain walks is not a square, that is not of the form \(x_1, \ldots, x_m, x_{1}, \ldots, x_m\) for any \(m \in \mathbb{N}\). Recently, various classes of walks have been suggested to be considered in the above definition. We construct graphs, for which the minimum number of colors needed for a square-free coloring is different if the considered set of walks vary, solving a problem posed by Brešar and Klavžar. We also prove the following: if an edge coloring of \(G\) is not square-free (even in the most general sense), then the length of the shortest square walk is at most \(8|E(G)|^2\). Hence, the necessary number of colors for a square-free coloring is algorithmically computable.
- Research article
- Full Text
- Ars Combinatoria
- Volume 087
- Pages: 359-375
- Published: 30/04/2008
If \(x\) is a vertex of a digraph \(D\), then we denote by \(d^+ (x)\) and \(d^- (x)\) the outdegree and the indegree of \(x\), respectively. The global irregularity of a digraph \(D\) is defined by \(i_g(D) = \max\{d^+ (x),d^- (x)\} – \min\{d^+ (y), d^- (y)\}\) over all vertices \(x\) and \(y\) of \(D\) (including \(x = y\)).
A \(c\)-partite tournament is an orientation of a complete \(c\)-partite graph. Recently, Volkmann and Winzen \([9]\) proved that \(c\)-partite tournaments with \(i_g(D) = 1\) and \(c \geq 3\) or \(i_g(D) = 2\) and \(c \geq 5\) contain a Hamiltonian path. Furthermore, they showed that these bounds are best possible.
Now, it is a natural question to generalize this problem by asking for the minimal value \(g(i,k)\) with \(i,k \geq 1\) arbitrary such that all \(c\)-partite tournaments \(D\) with \(i_g(D) \leq i\) and \(c \geq g(i,k)\) have a path covering number \(pc(D) \leq k\). In this paper, we will prove that \(4i-4k \leq g(i,k) \leq 4i-3k-1\), when \(i \geq k+2\). Especially in the case \(k = 1\), this yields that \(g(i, 1) = 4i-4\), which means that all \(c\)-partite tournaments \(D\) with the global irregularity \(i_g(D) = i\) and \(c \geq 4i-4\) contain a Hamiltonian path.
- Research article
- Full Text
- Ars Combinatoria
- Volume 087
- Pages: 353-358
- Published: 30/04/2008
In this paper, we discuss a problem on packing a unit cube with smaller cubes, which is a generalization of one of Erdős’ favorite problems: the square-packing problem. We first give the definition of the packing function \(f_3(n)\), then give the bounds for \(f_3(n)\).