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
- https://doi.org/10.61091/ars158-02
- Full Text
- Ars Combinatoria
- Volume 158
- Pages: 11-17
- Published: 31/03/2024
In this paper, we investigate the \((d,1)\)-total labelling of generalized Petersen graphs \(P(n,k)\) for \(d\geq 3\). We find that the \((d,1)\)-total number of \(P(n,k)\) with \(d\geq 3\) is \(d+3\) for even \(n\) and odd \(k\) or even \(n\) and \(k=\frac{n}{2}\), and \(d+4\) for all other cases.
- Research article
- https://www.doi.org/10.61091/ars158-01
- Full Text
- Ars Combinatoria
- Volume 158
- Pages: 3-10
- Published: 31/03/2024
By employing Kummer and Thomae transformations, we examine four classes of nonterminating \(_3F_2\)(1)-series with five integer parameters. Several new summation formulae are established in closed form.
- Research article
- https://dx.doi.org/10.61091/ars157-14
- Full Text
- Ars Combinatoria
- Volume 157
- Pages: 143-157
- Published: 25/12/2023
Let \(G=(V,\,E)\) be a simple graph with vertex set \(V(G)\) and edge set \(E(G)\). The Lanzhou index of a graph \(G\) is defined by \(Lz(G)=\sum\limits_{u \in V(G)} d_u^2\overline{d}_u\), where \(d_u\) (\(\overline{d}_u \) resp.) denotes the degree of the vertex \(u\) in \(G\) (\(\overline{G}\), the complement graph of \(G\) resp.). It has predictive powers to provide insights of chemical relevant properties of chemical graph structures. In this paper we discuss some properties of Lanzhou index. Several inequalities having lower and upper bound for the Lanzhou index in terms of first, second and third Zagreb indices, radius of graph, eccentric connectivity index, Schultz index, inverse sum indeg index and symmetric division deg index, are discussed. At the end the Lanzhou index of corona and join of graphs have been derived.
- Research article
- https://dx.doi.org/10.61091/ars157-13
- Full Text
- Ars Combinatoria
- Volume 157
- Pages: 133-141
- Published: 25/12/2023
We define an extremal \((r|\chi)\)-graph as an \(r\)-regular graph with chromatic number \(\chi\) of minimum order. We show that the Turán graphs \(T_{ak,k}\), the antihole graphs and the graphs \(K_k\times K_2\) are extremal in this sense. We also study extremal Cayley \((r|\chi)\)-graphs and we exhibit several \((r|\chi)\)-graph constructions arising from Turán graphs.
- Research article
- https://dx.doi.org/10.61091/ars157-12
- Full Text
- Ars Combinatoria
- Volume 157
- Pages: 121-131
- Published: 25/12/2023
A dominating broadcast of a graph \(G\) is a function \(f : V(G) \rightarrow \lbrace 0, 1, 2, \dots ,\text{diam}(G)\rbrace\) such that \(f(v) \leqslant e(v)\) for all \(v \in V(G)\), where \(e(v)\) is the eccentricity of \(v\), and for every vertex \(u \in V(G)\), there exists a vertex \(v\) with \(f(v) > 0\) and \(\text{d}(u,v) \leqslant f(v)\). The cost of \(f\) is \(\sum_{v \in V(G)} f(v)\). The minimum of costs over all the dominating broadcasts of \(G\) is called the broadcast domination number \(\gamma_{b}(G)\) of \(G\). A graph $G$ is said to be radial if \(\gamma_{b}(G)=\text{rad}(G)\). In this article, we give tight upper and lower bounds for the broadcast domination number of the line graph \(L(G)\) of \(G\), in terms of \(\gamma_{b}(G)\), and improve the upper bound of the same for the line graphs of trees. We present a necessary and sufficient condition for radial line graphs of central trees, and exhibit constructions of infinitely many central trees \(T\) for which \(L(T)\) is radial. We give a characterization for radial line graphs of trees, and show that the line graphs of the \(i\)-subdivision graph of \(K_{1,n}\) and a subclass of caterpillars are radial. Also, we show that \(\gamma_{b}(L(C))=\gamma(L(C))\) for any caterpillar \(C\).
- Research article
- https://dx.doi.org/10.61091/ars157-11
- Full Text
- Ars Combinatoria
- Volume 157
- Pages: 109-120
- Published: 25/12/2023
In this paper we introduce the concept of independent fixed connected geodetic number and investigate its behaviours on some standard graphs. Lower and upper bounds are found for the above number and we characterize the suitable graphs achieving these bounds. We also define two new parameters connected geo-independent number and upper connected geo-independent number of a graph. Few characterization and realization results are formulated for the new parameters. Finally an open problem is posed.
- Research article
- https://dx.doi.org/10.61091/ars157-10
- Full Text
- Ars Combinatoria
- Volume 157
- Pages: 95-108
- Published: 25/12/2023
Let \(E(H)\) and \(V(H)\) denote the edge set and the vertex set of the simple connected graph \(H\), respectively. The mixed metric dimension of the graph \(H\) is the graph invariant, which is the mixture of two important graph parameters, the edge metric dimension and the metric dimension. In this article, we compute the mixed metric dimension for the two families of the plane graphs viz., the Web graph \(\mathbb{W}_{n}\) and the Prism allied graph \(\mathbb{D}_{n}^{t}\). We show that the mixed metric dimension is non-constant unbounded for these two families of the plane graph. Moreover, for the Web graph \(\mathbb{W}_{n}\) and the Prism allied graph \(\mathbb{D}_{n}^{t}\), we unveil that the mixed metric basis set \(M_{G}^{m}\) is independent.
- Research article
- https://dx.doi.org/10.61091/ars157-09
- Full Text
- Ars Combinatoria
- Volume 157
- Pages: 89-94
- Published: 25/12/2023
Consider a total labeling \(\xi\) of a graph \(G\). For every two different edges \(e\) and \(f\) of \(G\), let \(wt(e) \neq wt(f)\) where weight of \(e = xy\) is defined as \(wt(e)=|\xi(e) – \xi(x) – \xi(y)|\). Then \(\xi\) is called edge irregular total absolute difference \(k\)-labeling of \(G\). Let \(k\) be the minimum integer for which there is a graph \(G\) with edge irregular total absolute difference labeling. This \(k\) is called the total absolute difference edge irregularity strength of the graph \(G\), denoted \(tades(G)\). We compute \(tades\) of \(SC_{n}\), disjoint union of grid and zigzag graph.
- Research article
- https://dx.doi.org/10.61091/ars157-08
- Full Text
- Ars Combinatoria
- Volume 157
- Pages: 81-88
- Published: 25/12/2023
A total dominator coloring of \(G\) without isolated vertex is a proper coloring of the vertices of \(G\) in which each vertex of \(G\) is adjacent to every vertex of some color class. The total dominator chromatic number \(\chi^t_d(G)\) of \(G\) is the minimum number of colors among all total dominator coloring of \(G\). In this paper, we will give the polynomial time algorithms to computing the total dominator coloring number for \(P_4\)-reducible and \(P_4\)-tidy graphs.
- Research article
- https://dx.doi.org/10.61091/ars157-07
- Full Text
- Ars Combinatoria
- Volume 157
- Pages: 73-80
- Published: 13/12/2023
An \(H\)-(a,d)-antimagic labeling in a \(H\)-decomposable graph \(G\) is a bijection \(f: V(G)\cup E(G)\rightarrow {\{1,2,…,p+q\}}\) such that \(\sum f(H_1),\sum f(H_2),\cdots, \sum f(H_h)\) forms an arithmetic progression with difference \(d\) and first element \(a\). \(f\) is said to be \(H\)-\(V\)-super-\((a,d)\)-antimagic if \(f(V(G))={\{1,2,…,p\}}\). Suppose that \(V(G)=U(G) \cup W(G)\) with \(|U(G)|=m\) and \(|W(G)|=n\). Then \(f\) is said to be \(H\)-\(V\)-super-strong-\((a,d)\)-antimagic labeling if \(f(U(G))={\{1,2,…,m\}}\) and \(f(W(G))={\{m+1,m+2,…,(m+n=p)\}}\). A graph that admits a \(H\)-\(V\)-super-strong-\((a,d)\)-antimagic labeling is called a \(H\)-\(V\)-super-strong-\((a,d)\)-antimagic decomposable graph. In this paper, we prove that complete bipartite graphs \(K_{m,n}\) are \(H\)-\(V\)-super-strong-\((a,d)\)-antimagic decomposable with both \(m\) and \(n\) are even.
- Research article
- Abstract Only
- Ars Combinatoria
- In Press
Sizhong Zhou\(^{1,*}\)
A graph \(G\) is called a fractional ID-\((g,f)\)-factor-critical covered graph if for any independent set \(I\) of \(G\) and for every edge \(e\in E(G-I)\), \(G-I\) has a fractional \((g,f)\)-factor \(h\) such that \(h(e)=1\). We give a sufficient condition using degree condition for a graph to be a fractional ID-\((g,f)\)-factor-critical covered graph. Our main result is an extension of Zhou, Bian and Wu’s previous result [S. Zhou, Q. Bian, J. Wu, A result on fractional ID-\(k\)-factor-critical graphs, Journal of Combinatorial Mathematics and Combinatorial Computing 87(2013)229–236] and Yashima’s previous result [T. Yashima, A
degree condition for graphs to be fractional ID-\([a,b]\)-factor-critical, Australasian Journal of Combinatorics 65(2016)191–199].