Cartesian-product networks combine well-studied graphs to create new structures with inherited properties, making them valuable for interconnection networks and parallel algorithms. Cycle decompositions of these networks are crucial for fault tolerance and adaptive routing. In this paper, we address the hypercube version of the Oberwolfach problem, focusing on decompositions of \(Q_n\) into cycles of equal or unequal lengths. We present an algorithm that enumerates all possible cycle types in \(Q_n\) and determine which decompositions are feasible or infeasible for \(Q_4\). Using an inductive approach, we extend these results to \(Q_n\) by leveraging distinct perfect matchings of \(Q_4\), yielding a variety of cycle decompositions. Additionally, we present results on factorizations of \(Q_n\) when \(n\) is a power of \(2\). These findings enhance the understanding of cycle structures in hypercubes and their applications to interconnection networks.

We investigate the combinatorial structure of edge-disjoint triangle packings in the complete graph \(K_n\). Two triangles are said to be edge-disjoint if they share no common edges, though they may share at most one vertex. For a given \(n\), let \(T_n\) denote the total number of subsets of triangles in \(K_n\) that are pairwise edge-disjoint, including the empty set, and let \(T_n^k\) denote the number of \(k\)-element sets of such triangles. In this article, we establish: (i) a general recurrence relation for \(T_n\) that enables asymptotic analysis, yielding the growth \(\log T_n = \Theta(n^2 \log n)\) for large \(n\); (ii) exact closed-form formulas for the number of edge-disjoint pairs (\(T_n^2\)), triples (\(T_n^3\)), and quadruples (\(T_n^4\)) of triangles in \(K_n\) for \(n \geq 6\). These results extend classical work on Steiner Triple Systems and provide new tools for analyzing triangle packings in complete graphs.

This paper introduces Hadamard-type \(t\)-Fibonacci-Lehmer (HTFL) sequences, a new hybrid construction combining Lehmer and Fibonacci recurrences. We establish their fundamental properties, including simple periodicity, and extend the definition to finite groups, with a detailed study of the Heisenberg group. Building on these results, we propose two Diffie–Hellman-style key exchange protocols based on upper-triangular unipotent matrices parameterized by HTFL sequence terms. Our work thus connects sequence theory, group theory, and cryptography in a novel way. While the algebraic framework and periodicity analysis are rigorous, we present the cryptographic constructions primarily as a conceptual foundation. We also discuss potential security considerations and outline directions for strengthening these schemes under formal hardness assumptions. This study demonstrates that HTFL sequences provide a fertile ground for both combinatorial investigations and future cryptographic applications.

A split graph is a graph in which the vertices can be partitioned into an independent set and a clique. We show that every nonsplit graph has at most four split maximal proper edge induced subgraphs. The exhaustive list of fifteen classes of nonsplit graphs having a split maximal proper edge induced subgraph is determined in this paper.

A kernel \(J\) of a digraph \(D\) is an independent set of vertices of \(D\) such that for every \(z\in V(D)\backslash J\) there exists an arc from \(z\) to \(J.\) A digraph \(D\) is said to be kernel-perfect if every induced subdigraph of it has a kernel. We characterise kernel-perfectness in special families of digraphs, namely, the line digraph, the subdivision digraph, the middle digraph, the digraph \(R(D)\) and the total digraph. We also obtain some results on kernel-perfectness in the generalised Mycielskian of digraphs. Moreover, we find some new classes of kernel-perfect digraphs by introducing a new product on digraphs.

The first, second Zagreb connection indices and modified first Zagreb connection index are defined as \(Z{C_1}(G)={\sum\limits_{{v\in V(G)}} {{\tau _G}^2(v)} }\), \(ZC{}_{2}(G)=\displaystyle\sum_{uv\in E(G)}^{}\tau{}_{G}(u)\tau{}_{G}(v)\) and \(ZC{}_{1}^{\ast }(G)=\displaystyle\sum_{v\in V(G)}^{}d{}_{G}(v)\tau{}_{G}(v)\), respectively. In this paper, we consider the maximum values of \(Z{C_1}(G)\), \(Z{C_2}(G)\), \(Z{C_1}^{*}(G)\) of \(n\)-vertex trees with fixed matching number \(m\) and the extremal graphs are also characterized.

An improper interval (edge) coloring of a graph \(G\) is an assignment of integer colors to the edges of \(G\) satisfying the condition that, for every vertex \(v \in V(G)\), the set of colors assigned to the edges incident with \(v\) forms an integral interval. An interval coloring is \(k\)-improper if at most \(k\) edges with the same color all share a common endpoint. The minimum integer \(k\) such that there exists a \(k\)-improper interval coloring of the graph \(G\) is the interval coloring impropriety of \(G\), denoted by \(\mu_{int}(G)\). In this paper, we provide a construction of an interval coloring of a subclass of complete multipartite graphs. Additionally, we determine improved upper bounds on the interval coloring impropriety of several classes of graphs, namely 2-trees, iterated triangulations, and outerplanar graphs. Finally, we investigate the interval coloring impropriety of the corona product of two graphs, \(G\odot H\).

A decomposition \(\mathcal{C}\) of a graph \(G\) is primitive if no proper, nontrivial subset of \(\mathcal{C}\) is a decomposition of an induced subgraph of \(G\). The existence of primitive decompositions has been studied for several decompositions, including path and cycle decompositions for complete and cocktail party graphs. In this work, we classify the existence of primitive star decompositions for complete graphs.