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.

Rajender Parsad1, Sanpei Kageyama2, V.K. Gupta1
1LA.S.R.L, Library Avenue, New Delhi — 110 012, India
2Hiroshima University, Higashi-Hiroshima 739-8524, Japan
Abstract:

Some results on combinatorial aspects of block designs using the complementary property have been obtained. The results pertain to non-existence of partially balanced incomplete block (PBIB) designs and identification of new \(2\)-associate and \(3\)-associate PBIB designs. A method of construction of extended group divisible (EGD) designs with three factors using self-complementary rectangular designs has also been given. Some rectangular designs have also been obtained using self-complementary balanced incomplete block designs. Catalogues of EGD designs and rectangular designs obtainable from these methods of construction, with number of replications \(\leq 10\) and block size \(\leq 10\) have been prepared.

Michael J.Ferrara1, Ronald J.Gould2, John R.Schmitt3
1 Department of Mathematics University of Colorado at Denver
2Department of Mathematics and Computer Science Emory University
3 Department of Mathematics Middlebury College
Abstract:

For any simple graph \(H\), let \(\sigma(H, n)\) be the minimum \(m\) so that for any realizable degree sequence \(\pi = (d_1, d_2, \ldots, d_n)\) with sum of degrees at least \(m\), there exists an \(n\)-vertex graph \(G\) witnessing \(\pi\) that contains \(H\) as a weak subgraph. Let \(F_{k}\) denote the friendship graph on \(2k+1\) vertices, that is, the graph of \(k\) triangles intersecting in a single vertex. In this paper, for \(n\) sufficiently large, \(\sigma(F_{k},n)\) is determined precisely.

Xianglin Wei1, Ren Ding1
1College of Mathematics, Hebei Normal University Shijiazhuang 050016, People’s Republic of China
Abstract:

Let \(C\) be a plane convex body, and let \(l(ab)\) be the Euclidean length of a longest chord of \(C\) parallel to the segment \(ab\) in \(C\). By the relative length of \(ab\) in a convex body \(C\), we mean the ratio of the Euclidean length of \(ab\) to \(\frac{l(ab)}{2}\). We say that a side \(ab\) of a convex \(n\)-gon is relatively short if the relative length of \(ab\) is not greater than the relative length of a side of the regular \(n\)-gon. In this article, we provide a significant sufficient condition for a convex hexagon to have a relatively short side.

David G.Glynn1, T.Aaron Gulliver2, Manish K.Gupta3
1School of Mathematical Sciences The University of Adelaide, SA 5005 Australia (previously Christchurch, New Zealand (Aotearoa))
2Department of Electrical & Computer Eng., University of Victoria, P.O. Box 3055, STN CSC, Victoria, B.C., Canada V8W 3P6
3Department of Mathematics & Statistics, Queens University 99 University Ave, Kingston, ON K7L 3N6, Canada
Abstract:

This paper studies families of self-orthogonal codes over \(\mathbb{Z}_4\). We show that the simplex codes (of Type \(a\) and Type \(\beta\)) are self-orthogonal. We answer the question of \(\mathbb{Z}_4\)-linearity for some codes obtained from projective planes of even order. A new family of self-orthogonal codes over \(\mathbb{Z}_4\) is constructed via projective planes of odd order. Properties such as self-orthogonality, weight distribution, etc. are studied. Finally, some self-orthogonal codes constructed from twistulant matrices are presented.

J.Richard Lundgren1, K. B.Reid2, Dustin J.Stewart1
1University of Colorado at Denver, Denver, CO 80217
2California State University San Marcos, San Marcos, CA 92096
Abstract:

A complete paired comparison digraph \(D\) is a directed graph in which \(xy\) is an arc for all vertices \(x,y\) in \(D\), and to each arc we assign a real number \(0 \leq a \leq 1\) called a weight such that if \(xy\) has weight \(a\) then \(yx\) has weight \(1 – a\). We say that two vertices \(x, y\) dominate a third \(z\) if the weights on \(xz\) and \(yz\) sum to at least \(1\). If \(x\) and \(y\) dominate all other vertices in a complete paired comparison digraph, then we say they are a dominant pair. We construct the domination graph of a complete paired comparison digraph \(D\) on the same vertices as \(D\) with an edge between \(x\) and \(y\) if \(x\) and \(y\) form a dominant pair in \(D\). In this paper, we characterize connected domination graphs of complete paired comparison digraphs. We also characterize the domination graphs of complete paired comparison digraphs with no arc weight of \(.5\).

Sabine Klinkenberg1, Lutz Volkmann1
1Lehrstuhl II fiir Mathematik, RWTH Aachen University, 52056 Aachen, Germany
Abstract:

A graph \(G\) is a \((d,d+k)\)-graph, if the degree of each vertex of \(G\) is between \(d\) and \(d+k\). Let \(p > 0\) and \(d+k \geq 2\) be integers. If \(G\) is a \((d,d+k)\)-graph of order \(n\) with at most \(p\) odd components and without a matching \(M\) of size \(2|M| = n – p\), then we show in this paper that

  1. \(n \geq 2d+p+2\) when \(p \leq k-2\),
  2. \(n \geq 2\left\lceil \frac{d(p+2)}{k} \right\rceil +p +2\) when \(p \geq k-1\).

Corresponding results for \(0 \leq p \leq 1\) and \(0 \leq k \leq 1\) were given by Wallis \([6]\), Zhao \([8]\), and Volkmann \([5]\).

Examples will show that the given bounds (i) and (ii) are best possible.

A. Elumalai1, G. Sethuraman1
1Department of Mathematics Crescent Engineering College, Chennai – 600 048
Abstract:

In this paper, we prove that the cycle \(C_n\) with parallel chords and the cycle \(C_n\) with parallel \(P_k\)-chords are cordial for any odd positive integer \(k \geq 3\) and for all \(n \geq 4\) except for \(n \equiv 4r + 2, r \geq 1\). Further, we show that every even-multiple subdivision of any graph \(G\) is cordial and we show that every graph is a subgraph of a cordial graph.

David Romero1, Abdon Sanchez-Arroyo2
1lInstituto de Mateméticas, Universidad Nacional Auténoma de México, Av. Univer- sidad s/n., 62210 Cuernavaca, Mor., Mexico.
2Calidad y Seguridad de la Informacién, Secretarfa de Hacienda y Crédito Publico, Constituyentes 1001-B, Piso 3, Col, Belén de las flores, 01110 México, D.F., Mexico.
Abstract:

A hypergraph is linear if no two distinct edges intersect in more than one vertex. A long standing conjecture of Erdős, Faber, and Lovász states that if a linear hypergraph has \(n\) edges, each of size \(n\), then its vertices can be properly colored with \(n\) colors. We prove the correctness of the conjecture for a new, infinite class of linear hypergraphs.

B. Balamohan1, R.B. Richter2
1University of Toronto
2University of Waterloo
Abstract:

We use a computer to show that the crossing number of generalized Petersen graph \(P(10,3)\) is six.

Peter Adams1, Darryn Bryant1, Mary Waterhouse1
1Department of Mathematics The University of Queensland Qld 4072 Australia
Abstract:

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

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