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.

Gil Kaplan1, Arieh Lev1, Yehuda Roditty1
1School of Computer Sciences The Academic College of Tel-Aviv-Yaffo 4 Antokolsky st., Tel-Aviv Israel 64044
Abstract:

In the first part of this paper, we present a generalization of complete graph factorizations obtained by labeling the graph vertices by natural numbers. In this generalization, the vertices are labeled by elements of an arbitrary group \(G\), in order to achieve a \(G\)-transitive factorization of the graph.

Lorenzo Milazzo1, Zsolt Tuza2,3
1Department of Mathematics, University of Catania, Viale A. Doria, 6 95125 – Catania, Italy.
2Department of Computer Science, H-8200 Veszprém, Egyetem u. 10, Hungary.
3Computer and Automation Institute, Hungarian Academy of Sciences, H-1111 Budapest, Kende u. 13-17;
Abstract:

Vertex colorings of Steiner systems \(S(t,t+1,v)\) are considered in which each block contains at least two vertices of the same color. Necessary conditions for the existence of such colorings with given parameters are determined, and an upper bound of the order \(O(\ln v)\) is found for the maximum number of colors. This bound remains valid for nearly complete partial Steiner systems, too. In striking contrast, systems \(S(t,k,v)\) with \(k \geq t+2\) always admit colorings with at least \(c\cdot v^\alpha\) colors, for some positive constants \(c\) and \(\alpha\), as \(v\to\infty\).

Jesse S.Beder1
1Department of Mathematics University of Wisconsin – Madison
Abstract:

Cwatsets were originally defined as subsets of \(\mathbb{Z}_2^d\) that are “closed with a twist.” Attempts have been made to generalize them, but the generalizations have failed to produce notions of subcwatset and quotient cwatset that behave naturally.

We present a new, abstract definition that appears to avoid these problems. The relationship between this new definition and its predecessor is similar to that between the abstract definition of “group” and its original meaning as a set of permutations. To justify the broader definition, we use small cancellation theory to prove a result analogous to the statement that every group is isomorphic to some permutation group. After developing the notion of a quotient cwatset, we prove an analogue of the First Homomorphism Theorem.

Gary E.Stevens1
1Department of Mathematics Hartwick College Oneonta, New York 13820 USA
Abstract:

In this paper, we consider a class of recursively defined, full binary trees called Lucas trees and investigate their basic properties. In particular, the distribution of leaves in the trees will be carefully studied. We then go on to show that these trees are \(2\)-splittable, i.e., they can be partitioned into two isomorphic subgraphs. Finally, we investigate the total path length and external path length in these trees, the Fibonacci trees, and other full \(m\)-ary trees.

Bing Yao1, Hui Cheng1, Ming Yao2, Meimei Zhao1
1College of Mathematics and Information Science, Northwest Normal University, Lanzhou, 730070, P.R.China
2Department of Information Process and Control Enginecring, Lanzhou Petrochemical College of Vocational Technology, 730060, P.R.China
Abstract:

A tree \(T\) with \(n\) vertices and a perfect matching \(M\) is strongly graceful if \(T\) admits a graceful labeling \(f\) such that \(f(u)+f(v) = n-1\) for every edge \(uv \in M\). Broersma and Hoede \([5]\) conjectured that every tree containing a perfect matching is strongly graceful in \(1999\). We prove that a tree \(T\) with diameter \(D(T) \leq 5\) supports the strongly graceful conjecture on trees. We show several classes of basic seeds and some constructive methods for constructing large scales of strongly graceful trees.

Yidong Sun1, Xiaoxia Wang2
1Department of Mathematics, Dalian Maritime University, 116026 Dalian, P.R. China
2Department of Mathematics, Shanghai University, 200444 Shanghai, P. R. China
Abstract:

In a previous paper, the first author introduced two classes of generalized Stirling numbers, \(s_m(n,k,p), S_m(n,k,p)\) with \(m = 1\) or \(2\), called \(p\)-Stirling numbers. In this paper, we discuss their determinant properties.

Jianxiu Hao1
1Institute of Mathematics, Physics and Information Sciences, Zhejiang Normal University, P. O. Box: 321004, Jinhua, Zhejiang, P.R. China
Abstract:

The Padmakar-Ivan (PI) index is a Wiener-Szeged-like topological index which reflects certain structural features of organic molecules. In this paper, we study the problem of PI index with respect to some simple pericondensed hexagonal systems and we solve it completely.

Yan Wang1
1Mathematics, Yan Tai University, Yan Tai 264005, China.
Abstract:

As a part of the author’s work of enumerating the edge-forwarding indices of Frobenius graphs, I give a class of valency four Frobenius graphs derived from the Frobenius groups \(\mathbb{Z}_{4n^2+1} \rtimes \mathbb{Z}_4\). Following the method of Fang, Li and Praeger, some properties including the diameter and the type of this class of graphs are given (Theorem \(3.2\)).

Alain C.Vandal1, Marston D.E.Conder2, Robert Gentleman3
1Department of Mathematics and Statistics, McGill University Centre for Clinical Epidemiology & Community Studies SMBD-Jewish General Hospital, Montréal
2Department of Mathematics, University of Auckland
3Fred Hutchison Cancer Research Center
Abstract:

We address the problem of determining all sets which form minimal covers of maximal cliques for interval graphs. We produce an algorithm enumerating all minimal covers using the C-minimal elements of the interval order, as well as an independence Metropolis sampler. We characterize maximal removable sets, which are the complements of minimal covers, and produce a distinct algorithm to enumerate them. We use this last characterization to provide bounds on the maximum number of minimal covers for an interval order with a given number of maximal cliques, and present some simulation results on the number of minimal covers in different settings.

Zihong Tian1
1Institute of Math., Hebei Normal University, Shijiazhuang 050016, P.R.China
Abstract:

A directed triple system of order \(v\), denoted by DTS\((v)\), is a pair \((X,\mathcal{B})\) where \(X\) is a \(v\)-set and \(\mathcal{B}\) is a collection of transitive triples on \(X\) such that every ordered pair of \(X\) belongs to exactly one triple of \(\mathcal{B}\). A DTS\((v)\) is called pure and denoted by PDTS\((v)\) if \((x,y,z) \in \mathcal{B}\) implies \((z,y,x) \notin \mathcal{B}\). A large set of disjoint PDTS\((v)\) is denoted by LPDTS\((v)\). In this paper, we establish the existence of LPDTS\((v)\) for \(v \equiv 0,4 \pmod{6}\), \(v\geq 4\).

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Special Issues

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;