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.

Meijie Ma1, Jun-Ming Xu2
1Department of Mathematics, Zhejiang Normal University Jinhua, 321004, China
2Department of Mathematics, University of Science and Technology of China Hefei, 230026, China
Abstract:

The locally twisted cube LTQn is a newly introduced interconnection network for parallel computing. As a variant of the hypercube Qn, LTQn has better properties than Qn with the same number of links and processors. Yang, Megson and Evans Evans [Locally twisted cubes are 4-pancyclic, Applied Mathematics Letters, 17(2004),919925] showed that LTQn contains a cycle of every length from 4 to 2n. In this note, we improve this result by showing that every edge of LTQn lies on a cycle of every length from 4 to 2n inclusive.

Haiyan Wang1, Yanxun Chang1
1Institute of Mathematics Beijing Jiaotong University Beijing 100044, P. R. China
Abstract:

Necessary and sufficient conditions are given for the existence of a (K3+e,λ)-group divisible design of type gtu1.

Nick C.Fiala1
1Department of Mathematics St. Cloud State University St. Cloud, MN 56301
Abstract:

A λ-design on v points is a set of v subsets (blocks) of a v-set such that any two distinct blocks meet in exactly λ points and not all of the blocks have the same size. Ryser’s and Woodall’s λ-design conjecture states that all 4-designs can be obtained from symmetric designs by a complementation procedure. In this paper, we establish feasibility criteria for the existence of λ-designs with two block sizes in the form of integrality conditions, equations, inequalities, and Diophantine equations involving various parameters of the designs. We use these criteria and a computer to prove that the λ-design conjecture is true for all λ-designs with two block sizes with v90 and λ45.

Emrah Kilic1, Dursun Tasci2
1TOBB Economics AND TECHNOLOGY UNIVERSITY MATHEMATICS DEPARTMENT 06560 ANKARA TURKEY
2D EPARTMENT OF MATHEMATICS, Gazi UNIVERSITY 06500 ANKARA TURKEY
Abstract:

In this paper, we consider the relationships between the sums of the Fibonacci and Lucas numbers and 1-factors of bipartite graphs.

Zoran Stojakovic1, Mila Stojakovic2
1Department of Mathematics and Informatics, Faculty of Science, University of Novi Sad 21000 Novi Sad, Serbia
2Department of Mathematics, Faculty of Engineering, University of Novi Sad 21000 Novi Sad, Serbia
Abstract:

We define extended orthogonal sets of d-cubes and show that they are equivalent to a class of orthogonal arrays, to geometric nets and a class of codes. As a corollary, an upper bound for the maximal number of d-cubes in an orthogonal set is obtained.

Yunging Zhang1
1Department of Mathematics, Nanjing University, Nanjing 210093, China
Abstract:

For two given graphs G1 and G2, the Ramseynumber R(G1,G2) is the smallest integer n such that for any graph G of order n, either G contains G1 or the complement of G contains G2. Let Pn denote a path of order n and Wm a wheel of order m+1. Chen et al. determined all values of R(Pn,Wm) for nm1. In this paper, we establish the best possible upper bound and determine some exact values for R(Pn,Wm) with nm2.

Bolian Liu1, Xiankun Zhang2
1Department of Mathematics South China Normal University Guangzhou,China
2Department of Mathematics West Virginia University Morgantown WV,U.S.A.
Abstract:

A container C(x,y) is a set of vertex-disjoint paths between vertices z and y in a graph G. The width w(C(x,y)) and length L(C(x,y)) are defined to be |C(x,y)| and the length of the longest path in C(x,y) respectively. The w-wide distance dw(x,y) between x and y is the minimum of L(C(x,y)) for all containers C(x,y) with width w. The w-wide diameter dw(G) of G is the maximum of dw(x,y) among all pairs of vertices x,y in G, xy. In this paper, we investigate some problems on the relations between dw(G) and diameter d(G) which were raised by D.F. Hsu [1]. Some results about graph equation of dw(G) are proved.

Manouchehr Zaker1
1 Institute for Advanced Studies in Basic Sciences 45195-1159, Zanjan – Iran
Abstract:

Greedy defining sets have been studied for the first time by the author for graphs. In this paper, we consider greedy defining sets for Latin squares and study the structure of these sets in Latin squares. We give a general bound for greedy defining numbers and linear bounds for greedy defining numbers of some infinite families of Latin squares. Greedy defining sets of circulant Latin squares are also discussed in the paper.

Xu Xirong1, Yang Yuansheng1, Li Huijun1, Xi Yue1
1Department of Computer Science Dalian University of Technolog Dalian, 116024, P. R. China
Abstract:

Let Cn(t) denote the cycle with n vertices, and Cn(t) denote the graphs consisting of t copies of Cn, with a vertex in common. Koh et al. conjectured that Cn(t) is graceful if and only if nt0,3(mod4). The conjecture has been shown true for n=3,5,6,7,9,4k. In this paper, the conjecture is shown to be true for n=11.

Xue-Feng Li1
1Department of Applied Mathematics and Physics Xi’an Institute of Post and Telecom Xi’ an 710121, China
Abstract:

Let P(G;λ) denote the chromatic polynomial of a graph G, expressed in the variable λ. Then G is said to be chromatically unique if G is isomorphic with H for any graph H such that P(H;λ)=P(G;λ). The graph consisting of s edge-disjoint paths joining two vertices is called an s-bridge graph. In this paper, we provide a new family of chromatically unique 5-bridge graphs.

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