We call \( T = (G_1, G_2, G_3) \) a graph-triple of order \( t \) if the \( G_i \) are pairwise non-isomorphic graphs on \( t \) non-isolated vertices whose edges can be combined to form \( K_t \). If \( m \geq t \), we say \( T \) divides \( K_m \) if \( E(K_m) \) can be partitioned into copies of the graphs in \( T \) with each \( G_i \) used at least once, and we call such a partition a \( T \)-multidecomposition. For each graph-triple \( T \) of order \( 6 \) for which it was not previously known, we determine all \( K_m \), \( m \geq 6 \), that admit a \( T \)-multidecomposition. Moreover, we determine maximum multipackings and minimum multicoverings when \( K_m \) does not admit a multidecomposition.

For the Firefighter Process with weights on the vertices, we show that the problem of deciding whether a subset of vertices of a total weight can be saved from burning remains NP-complete when restricted to binary trees. In addition, we show that a greedy algorithm that defends the vertex of highest degree adjacent to a burning vertex is not an \(\epsilon\)-\emph{approximation} algorithm for any \(\epsilon \in (0, 1]\) for the problem of determining the maximum weight that can be saved. This closes an open problem posed by MacGillivray and Wang.

Let \( K_v \) be the complete graph with \( v \) vertices. Let \( G \) be a finite simple graph. A \( G \)-decomposition of \( K_v \), denoted by \((v, G, 1)\)-GD, is a pair \((X, \mathcal{B})\), where \( X \) is the vertex set of \( K_v \), and \(\mathcal{B}\) is a collection of subgraphs of \( K_v \), called blocks, such that each block is isomorphic to \( G \). In this paper, the discussed graphs are \( G_i \), \( i = 1, 2, 3, 4 \), where \( G_i \) are four kinds of graphs with eight vertices and eight edges. We obtain the existence spectrum of \((v, G_i, 1)\)-GD.

We present a simple bijection between the set of triangulations of a convex polygon and the set of \(312\)-avoiding permutations.

A \emph{2-rainbow dominating function} of a graph \( G \) is a function \( g \) that assigns to each vertex a set of colors chosen from the set \( \{1, 2\} \) so that for each vertex \( v \) with \( g(v) = \emptyset \) we have \( \cup_{u \in N(v)} g(u) = \{1, 2\} \). The minimum of \( g(V(G)) = \sum_{v \in V(G)} |g(v)| \) over all such functions is called the \emph{2-rainbow domination number} \( \gamma_{r2}(G) \). A 2-rainbow dominating function \( g \) of a graph \( G \) is independent if no two vertices assigned non-empty sets are adjacent. The \emph{independent 2-rainbow domination number} \( i_{r2}(G) \) is the minimum weight of an independent 2-rainbow dominating function of \( G \). In this paper, we study independent 2-rainbow domination in graphs. We present some bounds and relations with other domination parameters.

For a connected graph \( G \) of order at least \( 3 \) and an integer \( k \geq 2 \), a \emph{twin edge} \( k \)-coloring of \( G \) is a proper edge coloring of \( G \) with the elements of \( \mathbb{Z}_k \), so that the induced vertex coloring in which the color of a vertex \( v \) in \( G \) is the sum (in \( \mathbb{Z}_k \)) of the colors of the edges incident with \( v \) is a proper vertex coloring. The minimum \( k \) for which \( G \) has a twin edge \( k \)-coloring is called the \emph{twin chromatic index} of \( G \) and is denoted by \( \chi_t'(G) \). It was conjectured that \( \Delta(T) \leq \chi_t'(T) \leq 2 + \Delta(T) \) for every tree of order at least \( 3 \), where \( \Delta(T) \) is the maximum degree of \( T \). This conjecture is verified for several classes of trees, namely brooms, double stars, and regular trees.

For a nontrivial connected graph \( G \), let \( c: V(G) \to \mathbb{Z}_2 \) be a vertex coloring of \( G \) where \( c(v) \neq 0 \) for at least one vertex \( v \) of \( G \). Then the coloring \( c \) induces a new coloring \( \sigma: V(G) \to \mathbb{Z}_2 \) of \( G \) defined by
\[
\sigma(v) = \sum_{u \in N[v]} c(u)
\]
where \( N[v] \) is the closed neighborhood of \( v \) and addition is performed in \( \mathbb{Z}_2 \). If \( \sigma(v) = 0 \in \mathbb{Z}_2 \) for every vertex \( v \) in \( G \), then the coloring \( c \) is called a (modular) monochromatic \( (2,0) \)-coloring of \( G \). A graph \( G \) having a monochromatic \( (2,0) \)-coloring is a (monochromatic) \( (2,0) \)-colorable graph. The minimum number of vertices colored \( 1 \) in a monochromatic \( (2,0) \)-coloring of \( G \) is the \( (2,0) \)-chromatic number of \( G \) and is denoted by \( \chi_{(2,0)}(G) \). For a \( (2,0) \)-colorable graph \( G \), the monochromatic \( (2,0) \)-spectrum \( S_{(2,0)}(G) \) of \( G \) is the set of all positive integers \( k \) for which exactly \( k \) vertices of \( G \) can be colored \( 1 \) in a monochromatic \( (2,0) \)-coloring of \( G \). Monochromatic \( (2,0) \)-spectra are determined for several well-known classes of graphs. If \( G \) is a connected graph of order \( n \geq 2 \) and \( a \in S_{(2,0)}(G) \), then \( a \) is even and \( 1 \leq |S_{(2,0)}(G)| \leq \left\lfloor \frac{n}{2} \right\rfloor \). It is shown that for every pair \( k,n \) of integers with \( 1 \leq k \leq \left\lfloor \frac{n}{2} \right\rfloor \), there is a connected graph \( G \) of order \( n \) such that \( |S_{(2,0)}(G)| = k \). A set \( S \) of positive even integers is \( (2,0) \)-realizable if \( S \) is the monochromatic \( (2,0) \)-spectrum of some connected graph. Although there are infinitely many non-\((2,0)\)-realizable sets, it is shown that every set of positive even integers is a subset of some \( (2,0) \)-realizable set. Other results and questions are also presented on \( (2,0) \)-realizable sets in graphs.

For two graphs \( H \) and \( G \), a decomposition \( \mathcal{D} = \{H_1, H_2, \ldots, H_k, R\} \) of \( G \) is called an \( H \)-maximal \( k \)-decomposition if \( H_i \cong H \) for \( 1 \leq i \leq k \) and \( R \) contains no subgraph isomorphic to \( H \). Let \(\text{Min}(G, H)\) and \(\text{Max}(G, H)\) be the minimum and maximum \( k \), respectively, for which \( G \) has an \( H \)-maximal \( k \)-decomposition. A graph \( G \) without isolated vertices is said to possess the intermediate decomposition property if for each connected graph \( G \) and each integer \( k \) with \(\text{Min}(G, H) \leq k \leq \text{Max}(G, H)\), there exists an \( H \)-maximal \( k \)-decomposition of \( G \). For a set \( S \) of graphs and a graph \( G \), a decomposition \( \mathcal{D} = \{H_1, H_2, \ldots, H_k, R\} \) of \( G \) is called an \( S \)-maximal \( k \)-decomposition if \( H_i \cong H \) for some \( H \in S \) for each integer \( i \) with \( 1 \leq i \leq k \) and \( R \) contains no subgraph isomorphic to any subgraph in \( S \). Let \(\text{Min}(G, S)\) and \(\text{Max}(G, S)\) be the minimum and maximum \( k \), respectively, for which \( G \) has an \( S \)-maximal \( k \)-decomposition. A set \( S \) of graphs without isolated vertices is said to possess the intermediate decomposition property if for every connected graph \( G \) and each integer \( k \) with \(\text{Min}(G, S) \leq k \leq \text{Max}(G, S)\), there exists an \( S \)-maximal \( k \)-decomposition of \( G \). While all those graphs of size \( 3 \) have been determined that possess the intermediate decomposition property, as have all sets consisting of two such graphs, here all remaining sets of graphs having size \( 3 \) that possess the intermediate decomposition property are determined.

An Eulerian graph \( G \) of size \( m \) is said to satisfy the Eulerian Cycle Decomposition Conjecture if the minimum number of odd cycles in a cycle decomposition of \( G \) is \( a \), the maximum number of odd cycles in a cycle decomposition is \( b \), and \( \ell \) is an integer such that \( a \leq \ell \leq b \) where \( \ell \) and \( m \) are of the same parity, then there is a cycle decomposition of \( G \) with exactly \( \ell \) odd cycles. Several regular complete \( 5 \)-partite graphs are shown to have this property.

An \( H_3 \) graph is a multigraph on three vertices with double edges between two pairs of distinct vertices and a single edge between the third pair. To settle the \( H_3 \) decomposition problem completely, one needs to complete the decomposition of a \( 2K_{10t+5} \) into \( H_3 \) graphs. In this paper, we present two new construction methods for such decompositions, resulting in previously unknown decompositions for \( v = 15, 25, 35, 45 \) and two new infinite families.