Brian Alspach1, Aditya Joshi1
1School of Information and Physical Sciences University of Newcastle Callaghan, NSW 2308 Australia
Abstract:

A graph \( X \) is \( k \)-spanning cyclable if for any subset \( S \) of \( k \) distinct vertices there is a 2-factor of \( X \) consisting of \( k \) cycles such that each vertex in \( S \) belongs to a distinct cycle. In this paper, we examine the \( k \)-spanning cyclability of 4-valent Cayley graphs on Abelian groups.

G.Princeton Lazarus1, K. Selvakumar2, P. Titus2
1Department of Science and Humanities, St.Mother Theresa Engineering College, Thoothukudi 628 102, India
2Department of Science and Humanities, University College of Engineering Nagercoil, Anna University: Tirunelveli Region, Nagercoil – 629 004, India
Abstract:

A path \(x_1, x_2, \dots, x_n\) in a connected graph \( G \) that has no edge \( x_i x_j \) \((j \geq i+3)\) is called a monophonic-triangular path or mt-path. A non-empty subset \( M \) of \( V(G) \) is a monophonic-triangular set or mt-set of \( G \) if every member in \( V(G) \) exists in a mt-path joining some pair of members in \( M \). The monophonic-triangular number or mt-number is the lowest cardinality of an mt-set of \( G \) and it is symbolized by \( mt(G) \). The general properties satisfied by mt-sets are discussed. Also, we establish \( mt \)-number boundaries and discover similar results for a few common graphs. Graphs \( G \) of order \( p \) with \( mt(G) = p \), \( p – 1 \), or \( p – 2 \) are characterized.

Youssef Ahendouz1, Ismail Akharraz1
1Mathematical and Informatics Engineering Laboratory Ibn Zohr University – Morocco
Abstract:

This note presents a counterexample to Propositions 7 and 8 in the paper [1], where the authors determine the values of \( V \) and \( W \). These values are crucial in determining the Hamming distance and MDS codes in the family of certain constacyclic codes over \(\mathbb{F}_{p^m}[u]/\langle u^3 \rangle\), which implies that the results found in [2] are incorrect. Furthermore, we provide corrections to the aforementioned results.

Arputha Jose. T.1, Sampath Kumar S.1, Cecily Sahai C.1
1Department of Mathematics, Sri Sivasubramaniya Nadar College of Engineering, Kalavakkam, Chennai – 603110, India
Abstract:

For a graph \( G \) and for non-negative integers \( p, q \) and \( r \), the triplet \( (p, q, r) \) is said to be an admissible triplet, if \( 3p + 4q + 6r = |E(G)| \). If \( G \) admits a decomposition into \( p \) cycles of length \( 3 \), \( q \) cycles of length \( 4 \), and \( r \) cycles of length \( 6 \) for every admissible triplet \( (p, q, r) \), then we say that \( G \) has a \( \{C_{3}^{p}, C_{4}^{q}, C_{6}^{r}\} \)-decomposition. In this paper, the necessary conditions for the existence of \( \{C_{3}^{p}, C_{4}^{q}, C_{6}^{r}\} \)-decomposition of \( K_{\ell, m, n}(\ell \leq m \leq n) \) are proved to be sufficient. This affirmatively answers the problem raised in \emph{Decomposing complete tripartite graphs into cycles of lengths \( 3 \) and \( 4 \), Discrete Math. 197/198 (1999), 123-135}. As a corollary, we deduce the main results of \emph{Decomposing complete tripartite graphs into cycles of lengths \( 3 \) and \( 4 \), Discrete Math., 197/198, 123-135 (1999)} and \emph{Decompositions of complete tripartite graphs into cycles of lengths \( 3 \) and \( 6 \), Austral. J. Combin., 73(1), 220-241 (2019)}.

Peter Adams1, Ryan C. Bunge2, Jordan Dulowski2, Saad I. El-Zanati2, Maddillon Kenney2, Ugur Odabası3, Kaitlyn Zale2
1The University of Queensland, QLD 4072, Australia
2Illinois State University, Normal, IL 61790-4520, USA
3Istanbul University-Cerrahpasa, Istanbul, 34320, Turkey
Abstract:

The λ-fold complete symmetric directed graph of order v, denoted λKv*, is the directed graph on v vertices and λ directed edges in each direction between each pair of vertices. For a given directed graph D, the set of all v for which λKv* admits a D-decomposition is called the λ-fold spectrum of D. In this paper, we settle the λ-fold spectrum of each of the nine non-isomorphic orientations of a 6-cycle.

Youssef Ahendouz1, Ismail Akharraz1
1Mathematical and Informatics Engineering Laboratory Ibn Zohr University – Morocco
Abstract:

In this paper, we provide a correction regarding the structure of negacyclic codes of length \(8p^s\) over \(\mathcal{R} = \mathbb{F}_{p^m} + u \mathbb{F}_{p^m}\) when \(p^m \equiv 3 \pmod{8}\) as classified in [1]. Among other results, we determine the number of codewords and the dual of each negacyclic code.

Xiaoling Sun1, Jianwei Du1
1School of Mathematics, North University of China, Taiyuan, Shanxi 030051, China
Abstract:

The multiplicative sum Zagreb index is a modified version of the well-known Zagreb indices. The multiplicative sum Zagreb index of a graph \(G\) is the product of the sums of the degrees of pairs of adjacent vertices. The mathematical properties of the multiplicative sum Zagreb index of graphs with given graph parameters deserve further study, as they can be used to detect chemical compounds and study network structures in mathematical chemistry. Therefore, in this paper, the maximal and minimal values of the multiplicative sum Zagreb indices of graphs with a given clique number are presented. Furthermore, the corresponding extremal graphs are characterized.

Kevin R. Hutson1, Stephen T. Hedetniemi2
1Department of Mathematics, Furman University, Greenville, SC 29613, USA
2Professor Emeritus of Computer Science, Clemson University, Clemson, SC 29634, USA
Abstract:

Let \( G = (V, E) \) be a graph. A subset \( S \subseteq V \) of vertices is an \textit{efficient dominating set} if every vertex \( v \in V \) is adjacent to exactly one vertex in \( S \), where a vertex \( u \in S \) is considered to be adjacent to itself. Efficient domination is highly desirable in many real-world applications, and yet, in general, graphs are often not efficient. It is of value, therefore, to determine optimum ways in which inefficient graphs can be changed in order to make them efficient. It is well known, for example, that almost no \( m \times n \) grid graphs have efficient dominating sets. In this paper, we consider the minimum number of vertices that can be removed from an \( m \times n \) grid graph so that the remaining graph has an efficient dominating set.

Yegnanarayanan Venkataraman1, George Barnabas1, Bryan Freyberg2
1Department of Mathematics, Kalasalingam Academy of Research and Education, Krishnankoil, 626126, Tamilnadu, India
2Department of Mathematics and Statistics, University of Minnesota Duluth, 1117 University Drive, Duluth, 55811, MN, USA
Abstract:

Let \( G = (V, E) \) be any graph. If there exists an injection \( f : V \rightarrow \mathbb{Z} \), such that \( |f(u) – f(v)| \) is prime for every \( uv \in E \), then we say \( G \) is a prime distance graph (PDG). The problem of characterizing the family of all prime distance graphs (PDGs) with chromatic number 3 or 4 is challenging. In the fourth part of this series of articles, we determined which fans are PDGs and which wheels are PDGs. In addition, we showed: (1) a chain of \( n \) mutually isomorphic PDGs is a PDG, and (2) the Cartesian product of a PDG and a path is a PDG. In this part of the series, we improve (1) by showing that there exists a chain of \( n \) arbitrary PDGs which is a PDG. We also show that the following graphs are PDGs: (a) any graph with at most three cycles, (b) the one-point union of cycles, and (c) a family of graphs consisting of paths with common end vertices.

M. Ilayaraja1, A. Muthusamy2
1Department of Mathematics Sona College of Arts and Science Salem-636005, Tamil Nadu, India
2Department of Mathematics Periyar University Salem-636011, Tamil Nadu, India
Abstract:

Let \(P_n\) and \(K_n\) respectively denote a path and complete graph on \(n\) vertices. By a \(\{pH_{1}, qH_{2}\}\)-decomposition of a graph \(G\), we mean a decomposition of \(G\) into \(p\) copies of \(H_{1}\) and \(q\) copies of \(H_{2}\) for any admissible pair of nonnegative integers \(p\) and \(q\), where \(H_{1}\) and \(H_{2}\) are subgraphs of \(G\). In this paper, we show that for any admissible pair of nonnegative integers \(p\) and \(q\), and positive integer \(n \geq 4\), there exists a \(\{pP_{4}, qS_{4}\}\)-decomposition of \(K_n\) if and only if \(3p+4q=\binom{n}{2}\), where \(S_4\) is a star with \(4\) edges.