Let \( M(b, n) \) be the complete multipartite graph with \( b \) parts \( B_0, \dots, B_{b-1} \) of size \( n \). A \( z \)-cycle system of \( M(b, n) \) is said to be a \({cycle-frame}\) if the \( z \)-cycles can be partitioned into sets \( S_1, \dots, S_k \) such that for \( 1 \leq j \leq k \), \( S_j \) induces a \( 2 \)-factor of \( M(b, n) \backslash B_i \) for some \( i \in \mathbb{Z}_b \). The existence of a \( C_z \)-frame of \( M(b, n) \) has been settled when \( z \in \{3, 4, 5, 6\} \). Here, we completely settle the case of \( C_z \)-frames when \( z \) is \( 8 \), and we give some solutions for larger values of \( z \).

A graph \( G \) is said to be a \( (2, k) \)-regular graph if each vertex of \( G \) is at a distance two away from \( k \) vertices of \( G \). A graph \( G \) is called an \( (r, 2, k) \)-regular graph if each vertex of \( G \) is at a distance \( 1 \) away from \( r \) vertices of \( G \) and each vertex of \( G \) is at a distance \( 2 \) away from \( k \) vertices of \( G \) \cite{9}. This paper suggests a method to construct a \( ((m + n – 2), 2, (m – 1)(n – 1)) \)-regular graph of smallest order \( mn \) containing a given graph \( G \) of order \( n \geq 2 \) as an induced subgraph for any \( m > 1 \).

A broadcast on a graph \( G \) is a function \( f: V \to \{0, 1, \dots, \text{diam}G\} \) such that \( f(v) < e(v) \) (the eccentricity of \( v \)) for all \( v \in V \). The broadcast number of \( G \) is the minimum value of \( \sum_{v \in V} f(v) \) among all broadcasts \( f \) for which each vertex of \( G \) is within distance \( f(v) \) from some vertex \( v \) with \( f(v) \geq 1 \). This number is bounded above by the radius of \( G \). A graph is uniquely radial if its only minimum broadcasts are broadcasts \( f \) such that \( f(v) = \text{rad}G \) for some central vertex \( v \), and \( f(u) = 0 \) if \( u \neq v \). We characterize uniquely radial trees.

In this paper, we refer to a decomposition of a tripartite graph into paths of length \( 3 \), or into \( 6 \)-cycles, as gregarious if each subgraph has at least one vertex in each of the three partite sets. For a tripartite graph to have a \( 6 \)-cycle decomposition it is straightforward to see that all three parts must have even size. Here we provide a gregarious decomposition of a complete tripartite graph \( K(r, s, t) \) into paths of length \( 3 \), and thus of \( K(2r, 2s, 2t) \) into gregarious \( 6 \)-cycles, in all possible cases, when the straightforward necessary conditions on \( r, s, t \) are satisfied.

For any graph \( G = (V, E) \), a non-empty set \( S \subseteq V \) is \({secure}\) if and only if \( |N[X] \cap S| \geq |N[X] – S| \) for all \( X \subseteq S \). The cardinality of a minimum secure set in \( G \) is the security number of \( G \). In this note, we give a new proof for the \({security\; number}\) of grid-like graphs.

Let \( G = (V, E) \) be a graph having at least \( 3 \) vertices in each of its components. A set \( L \subseteq V(G) \) is a liar’s dominating set if

1. \( |N_G[v] \cap L| \geq 2 \) for all \( v \in V(G) \) and
2. \( |(N_G[u] \cup N_G[v]) \cap L| \geq 3 \) for every pair \( u, v \in V(G) \) of distinct vertices in \( G \),

where \( N_G[x] = \{y \in V \mid xy \in E\} \cup \{x\} \) is the closed neighborhood of \( x \) in \( G \). In this paper, we characterize the vertices that are contained in all or in no minimum liar’s dominating sets in trees. Given a tree \( T \), we also propose a polynomial time algorithm to compute the set of all vertices which are contained in every minimum liar’s dominating set of \( T \) and the set of all vertices which are not contained in any minimum liar’s dominating set of \( T \).

A graph is chordal if and only if every cycle either has a chord or is a triangle. If an edge (or triangle) is defined to be a strength-\(k\) edge (or triangle) whenever it is in at least \( k \) maximum cliques, then a graph is strongly chordal if and only if, for every \( k \geq 1 \), every cycle of strength-\(k\) edges either has a strength-\(k\) chord or is a strength-\(k\) triangle. Dual-chordal graphs have been defined so as to be the natural cycle/cutset duals of chordal graphs. A carefully crafted notion of dual strength allows a characterization of strongly dual-chordal graphs that is parallel to the above. This leads to a complete list of all \( 3 \)-connected strongly dual-chordal graphs.

An edge-coloured path is rainbow if the colours of its edges are distinct. For a positive integer \( k \), an edge-colouring of a graph \( G \) is rainbow \( k \)-connected if any two vertices of \( G \) are connected by \( k \) internally vertex-disjoint rainbow paths. The rainbow \( k \)-connection number \( rc_k(G) \) is defined to be the minimum integer \( t \) such that there exists an edge-colouring of \( G \) with \( t \) colours which is rainbow \( k \)-connected. We consider \( rc_2(G) \) when \( G \) has fixed vertex-connectivity. We also consider \( rc_k(G) \) for large complete bipartite and multipartite graphs \( G \) with equipartitions. Finally, we determine sharp threshold functions for the properties \( rc_k(G) = 2 \) and \( rc_k(G) = 3 \), where \( G \) is a random graph. Related open problems are posed.

A Costas array of order \(n\) is an \(n \times n\) permutation matrix with the property that all of the \(n(n-1)/2\) line segments between pairs of \(1\)’s differ in length or in slope. A Costas latin square of order \(n\) is an \(n \times n\) latin square where for each symbol \(k\), with \(1 \leq k \leq n\), the cells containing \(k\) determine a Costas array. The existence of a Costas latin square of side \(n\) is equivalent to the existence of \(n\) mutually disjoint Costas arrays. In 2012, Dinitz, Östergird, and Stinson enumerated all Costas latin squares of side \(n \leq 27\). In this brief note, a sequel to that paper, we extend this search to sides \(n = 28\) and \(29\). In addition, we determine the sizes of maximal sets of disjoint Costas latin squares of side \(n\) for \(n \leq 29\).

For a given graph \( G \), the set of positive integers \( v \) for which a \( G \)-design exists is usually called the spectrum for \( G \) and the determination of the spectrum is sometimes called the spectrum problem. We consider the spectrum problem for \( G \)-designs satisfying additional conditions of balance, in the case where \( G \) is a member of one of the following infinite families of trees: caterpillars, stars, comets, lobsters, and trees of diameter at most \( 5 \). We determine the existence spectrum for balanced \( G \)-designs, degree-balanced and partially degree-balanced \( G \)-designs, and orbit-balanced \( G \)-designs. We also address the existence question for non-balanced \( G \)-designs, for \( G \)-designs which are either balanced or partially degree-balanced but not degree-balanced, and for \( G \)-designs which are degree-balanced but not orbit-balanced.