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.

V. Sitaramaiah1, M. V.Subbarao2
1 Department of Mathematics, Pondicherry Engineering College, Pondicherry, 605 014, India.
2Department of Mathematical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G2G1.
Emrah Kilic1
1TOBB University oF Ecoxomics AND TECHNOLOGY, MATHEMATICS DEPARTMENT 06560 S6acTozl, ANKARATURKEY
Abstract:

In this paper, we consider a certain second order linear recurrence and then give generating matriees for the sums of positively and negatively subscripted terms of this recurrence. Further, we use matrix methods and derive explicit. formulas for these sums.

Dieter Rautenbach1
1Forschungsinstitut ftir Diskrete Mathematik Lennéstr. 2, D-53113 Bonn, Germany
Abstract:

For a simple and finite graph \(G = (V,E)\), let \(w_{\max}(G)\) be the maximum total weight \(w(E) = \sum_{e\in E} w(e)\) of \(G\) over all weight functions \(w: E \to \{-1,1\}\) such that \(G\) has no positive cut, i.e., all cuts \(C\) satisfy \(w(C) \leq 0\).

For \(r \geq 1\), we prove that \(w_{\max}(G) \leq -\frac{|V|}{2}\) if \(G\) is \((2r-1)\)-regular and \(w_{\max}(G) \leq -\frac{r|V|}{2r+1}\) if \(G\) is \(2r\)-regular. We conjecture the existence of a constant \(c\) such that \(w_{\max}(G) \leq -\frac{5|V|}{6} + c\) if \(G\) is a connected cubic graph and prove a special case of this conjecture. Furthermore, as a weakened version of this conjecture, we prove that \(w_{\max}(G) \leq -\frac{2|V|}{3}+\frac{2}{3}\) if \(G\) is a connected cubic graph.

Sun Yongqi1, Yang Yuansheng1, Lin Xiaohui1, Zheng Wenping2
1Department of Computer Science, Dalian University of Technology Dalian, 116024, P. R. China
2Department of Computer Science, Dalian University of Technology Dalian, 116024, P. R. ChinaZheng Wenping
Abstract:

Let \(G_i\) be the subgraph of \(G\) whose edges are in the \(i\)-th color in an \(r\)-coloring of the edges of \(G\). If there exists an \(r\)-coloring of the edges of \(G\) such that \(H_i \nsubseteq G_i\) for all \(1 \leq i \leq r\), then \(G\) is said to be \(r\)-colorable to \((H_1, H_2, \ldots, H_r)\). The multicolor Ramsey number \(R(H_1, H_2, \ldots, H_r)\) is the smallest integer \(n\) such that \(K_n\) is not \(r\)-colorable to \((H_1, H_2, \ldots, H_r)\). It is well known that \(R(C_m, C_4, C_4) = m + 2\) for sufficiently large \(m\). In this paper, we determine the values of \(R(C_m, C_4, C_4)\) for \(m \geq 5\), which show that \(R(C_m, C_4, C_4) = m + 2\) for \(m \geq 11\).

Andrea Vietri1
1Dipartimento Me.Mo.Mat., via A. Scarpa 16, 00161 Rome, Italy.
Abstract:

The proof of gracefulness for the Generalised Petersen Graph \(P_{8t,3}\) for every \(t \geq 1\), written by the same author (Graceful labellings for an infinite class of generalised Petersen graphs, Ars Combinatoria \(81 (2006)\), pp. \(247-255)\), requires the change of just one label, for the only case \(t = 5\).

Arnold Knopfmacher1, Helmut Prodinger2
1THE JOHN KNOPFMACHER CENTRE FOR APPLICABLE ANAL- YSIS AND NUMBER THEORY, UNIVERSITY OF THE WITWATERSRAND, P. O. Wits, 2050 JOHANNESBURG, SOUTH AFRICA
2THE JOHN KNOPFMACHER CENTRE FOR APPLICABLE ANALYSIS AND NUMBER THEORY, DEPARTMENT OF MATHEMATICS, UNIVERSITY OF THE WITWATER- SRAND, P. O. WiTs, 2050 JOHANNESBURG, SOUTH AFRICA
Abstract:

For words of length \(n\), generated by independent geometric random variables, we study the average initial and end heights of the last descent in the word. In addition, we compute the average initial and end height of the last descent in a random permutation of \(n\) letters.

Yeow Meng Chee1,2
1Interactive Digital Media Program Office Media Development Authority 140 Hill Street Singapore 179369
2Division of Mathematical Sciences School of Physical and Mathematical Sciences Nanyang Technological University Singapore 637616
Abstract:

We construct a record-breaking binary code of length \(17\), minimal distance \(6\), constant weight \(6\), and containing \(113\) codewords.

Zhizheng Zhang 1, Xiaoli Ye1
1Department of Mathematics, Luoyang Teachers’ College, Luoyang 471022, P. R. China
Abstract:

The purpose of this note is to give the power formula of the generalized Lah matrix and show \(\mathcal{L}[x,y] = \mathcal{FQ}[x,y]\), where \(\mathcal{F}\) is the Fibonacci matrix and \(\mathcal{Q}[x,y]\) is the lower triangular matrix. From it, several combinatorial identities involving the Fibonacci numbers are obtained.

S. Ramachandran1, S. Monikandan1
1Department of Mathematics, Vivekananda College, Agasieeswaram-629 701, Kanyakumari, T.N. State, INDIA.
Abstract:

A graph is called set reconstructible if it is determined uniquely (up to isomorphism) by the set of its vertex-deleted subgraphs. We prove that some classes of separable graphs with a unique endvertex are set reconstructible and show that all graphs are set reconstructible if all \(2\)-connected graphs are set reconstructible.

Arie Bialostocki1, David J.Grynkiewicz2
1300 Brink Hall, University of Idaho, P.O. Box 441103, Moscow, ID 83844-1103,
2Mathematics 253-37, Caltech, Pasadena, CA 91125
Abstract:

We prove the following extension of the Erdős-Ginzburg-Ziv Theorem. Let \(m\) be a positive integer. For every sequence \(\{a_i\}_{i\in I}\) of elements from the cyclic group \(\mathbb{Z}_m\), where \(|I| = 4m – 5\) (where \(|I| = 4m – 3\)), there exist two subsets \(A, B \subseteq I\) such that \(|A \cap B| = 2\) (such that \(|A \cap B| = 1\)), \(|A| = |B| = m\), and \(\sum\limits_{i\in b} a_i = \sum\limits_{i\in b} b_i = 0\).

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The Combinatorial Press Editorial Office routinely extends invitations to scholars for the guest editing of Special Issues, focusing on topics of interest to the scientific community. We actively encourage proposals from our readers and authors, directly submitted to us, encompassing subjects within their respective fields of expertise. The Editorial Team, in conjunction with the Editor-in-Chief, will supervise the appointment of Guest Editors and scrutinize Special Issue proposals to ensure content relevance and appropriateness for the journal. To propose a Special Issue, kindly complete all required information for submission;