Necessary conditions for the existence of a \( 3 \)-GDD\((n, 3, k; \lambda_1, \lambda_2)\) are obtained along with some non-existence results. We also prove that these necessary conditions are sufficient for the existence of a \( 3 \)-GDD\((n, 3, 4; \lambda_1, \lambda_2)\) for \( n \) even.

The paired domination subdivision number \( \text{sd}_{pr}(G) \) of a graph \( G \) is the minimum number of edges that must be subdivided (where an edge can be subdivided at most once) in order to increase the paired domination number of \( G \). We prove that the decision problem of the paired domination subdivision number is NP-complete even for bipartite graphs. For this reason, we define the \textit{paired domination multisubdivision number} of a nonempty graph \( G \), denoted by \( \text{msd}_{pr}(G) \), as the minimum positive integer \( k \) such that there exists an edge which must be subdivided \( k \) times to increase the paired domination number of \( G \). We show that \( \text{msd}_{pr}(G) \leq 4 \) for any graph \( G \) with at least one edge. We also determine paired domination multisubdivision numbers for some classes of graphs. Moreover, we give a constructive characterization of all trees with paired domination multisubdivision number equal to 4.

Consider the multigraph obtained by adding a double edge to \( K_4 – e \). Now, let \( D \) be a directed graph obtained by orientating the edges of that multigraph. We establish necessary and sufficient conditions on \( n \) for the existence of a \( (K^*_n, D) \)-design for four such orientations.

Working on general hypergraphs requires to properly redefine the concept of adjacency in a way that it captures the information of the hyperedges independently of their size. Coming to represent this information in a tensor imposes to go through a uniformisation process of the hypergraph. Hypergraphs limit the way of achieving it as redundancy is not permitted. Hence, our introduction of hb-graphs, families of multisets on a common universe corresponding to the vertex set, that we present in details in this article, allowing us to have a construction of adequate adjacency tensor that is interpretable in term of \(m\)-uniformisation of a general hb-graph. As hypergraphs appear as particular hb-graphs, we deduce two new (\(e\))-adjacency tensors for general hypergraphs. We conclude this article by giving some first results on hypergraph spectral analysis of these tensors and a comparison with the existing tensors for general hypergraphs, before making a final choice.

The paper proposes techniques which provide closed-form solutions for special simultaneous systems of two and three linear recurrences. These systems are characterized by particular restrictions on their coefficients. We discuss the application of these systems to some algorithmic problems associated with the relationship between algebraic expressions and graphs. Using decomposition methods described in the paper, we generate the simultaneous recurrences for graph expression lengths and solve them with the proposed approach.

For a given graph \(G\), a variation of its line graph is the 3-xline graph, where two 3-paths \(P\) and \(Q\) are adjacent in \(G\) if \(V(P) \cap V(Q) = \{v\}\) when \(v\) is the interior vertex of both \(P\) and \(Q\). The vertices of the 3-xline graph \(XL_3(G)\) correspond to the 3-paths in \(G\), and two distinct vertices of the 3-xline graph are adjacent if and only if they are adjacent 3-paths in \(G\). In this paper, we study 3-xline graphs for several classes of graphs, and show that for a connected graph \(G\), the 3-xline graph is never isomorphic to \(G\) and is connected only when \(G\) is the star \(K_{1,n}\) for \(n = 2\) or \(n \geq 5\). We also investigate cycles in 3-xline graphs and characterize those graphs \(G\) where \(XL_3(G)\) is Eulerian.

An \(L(h,k)\) labeling of a graph \(G\) is an integer labeling of the vertices where the labels of adjacent vertices differ by at least \(h\), and the labels of vertices that are at distance two from each other differ by at least \(k\). The span of an \(L(h,k)\) labeling \(f\) on a graph \(G\) is the largest label minus the smallest label under \(f\). The \(L(h,k)\) span of a graph \(G\), denoted \(\lambda_{h,k}(G)\), is the minimum span of all \(L(h,k)\) labelings of \(G\).

In this paper, we use standard graph labeling techniques to prove that each tri-cyclic graph with eight edges decomposes the complete graph \(K_n\) if and only if \(n \equiv 0, 1 \pmod{16}\). We apply \(\rho\)-tripartite labelings and 1-rotational \(\rho\)-tripartite labelings.

We introduce a variation of \(\sigma\)-labeling to prove that every disconnected unicyclic bipartite graph with eight edges decomposes the complete graph \(K_n\) whenever the necessary conditions are satisfied. We combine this result with known results in the connected case to prove that every unicyclic bipartite graph with eight edges other than \(C_8\) decomposes \(K_n\) if and only if \(n \equiv 0, 1 \pmod{16}\) and \(n > 16\).