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In this paper, we propose and investigate the concept of $k$-coalitions in graphs, where $k\ge 1$ is an integer. A $k$-coalition refers to a pair of disjoint vertex sets that jointly constitute a $k$-dominating set of the graph, meaning that every vertex not in the set has at least $k$ neighbors in the set. We define a $k$-coalition partition of a graph as a vertex partition in which each set is either a $k$-dominating set with exactly $k$ members or forms a $k$-coalition with another set in the partition. The maximum number of sets in a $k$-coalition partition is called the $k$-coalition number of the graph represented by $C_k(G)$. We present fundamental findings regarding the properties of $k$-coalitions and their connections with other graph parameters. We obtain the exact values of $2$-coalition number of some specific graphs and also study graphs with large $2$-coalition number., Comment: 13 pages, 1 figure

A subset of vertices $S$ of a graph $G$ is a dominating set if every vertex in $V \setminus S$ has at least one neighbor in $S$. A domatic partition is a partition of the vertices of a graph $G$ into disjoint dominating sets. The domatic number $d(G)$ is the maximum size of a domatic partition. Suppose that $dp(G,i)$ is the number of distinct domatic partition of $G$ with cardinality $i$. In this paper, we consider the generating function of $dp(G,i)$, i.e., $DP(G,x)=\sum_{i=1}^{d(G)}dp(G,i)x^i$ which we call it the domatic partition polynomial. We explore the domatic polynomial for trees, providing a quadratic time algorithm for its computation based on weak 2-coloring numbers. Our results include specific findings for paths and certain graph products, demonstrating practical applications of our theoretical framework.

A set $S$ of vertices in a graph $G$ is a dominating set if every vertex of $V(G) \setminus S$ is adjacent to a vertex in $S$. A coalition in $G$ consists of two disjoint sets of vertices $X$ and $Y$ of $G$, neither of which is a dominating set but whose union $X \cup Y$ is a dominating set of $G$. Such sets $X$ and $Y$ form a coalition in $G$. A coalition partition, abbreviated $c$-partition, in $G$ is a partition $\mathcal{X} = \{X_1,\ldots,X_k\}$ of the vertex set $V(G)$ of $G$ such that for all $i \in [k]$, each set $X_i \in \mathcal{X}$ satisfies one of the following two conditions: (1) $X_i$ is a dominating set of $G$ with a single vertex, or (2) $X_i$ forms a coalition with some other set $X_j \in \mathcal{X}$. %The coalition number ${C}(G)$ is the maximum cardinality of a $c$-partition of $G$. Let ${\cal A} = \{A_1,\ldots,A_r\}$ and ${\cal B}= \{B_1,\ldots, B_s\}$ be two partitions of $V(G)$. Partition ${\cal B}$ is a refinement of partition ${\cal A}$ if every set $B_i \in {\cal B} $ is either equal to, or a proper subset of, some set $A_j \in {\cal A}$. Further if ${\cal A} \ne {\cal B}$, then ${\cal B}$ is a proper refinement of ${\cal A}$. Partition ${\cal A}$ is a minimal $c$-partition if it is not a proper refinement of another $c$-partition. Haynes et al. [AKCE Int. J. Graphs Combin. 17 (2020), no. 2, 653--659] defined the minmin coalition number $c_{\min}(G)$ of $G$ to equal the minimum order of a minimal $c$-partition of $G$. We show that $2 \le c_{\min}(G) \le n$, and we characterize graphs $G$ of order $n$ satisfying $c_{\min}(G) = n$. A polynomial-time algorithm is given to determine if $c_{\min}(G)=2$ for a given graph $G$. A necessary and sufficient condition for a graph $G$ to satisfy $c_{\min}(G) \ge 3$ is given, and a characterization of graphs $G$ with minimum degree~$2$ and $c_{\min}(G)= 4$ is provided.

Let $G$ be graph with vertex set $V$ and order $n=|V|$. A coalition in $G$ is a combination of two distinct sets, $A\subseteq V$ and $B\subseteq V$, which are disjoint and are not dominating sets of $G$, but $A\cup B$ is a dominating set of $G$. A coalition partition of $G$ is a partition $\mathcal{P}=\{S_1,\ldots,S_k\}$ of its vertex set $V$, where each set $S_i\in \mathcal{P}$ is either a dominating set of $G$ with only one vertex, or it is not a dominating set but forms a coalition with some other set $S_j \in \mathcal{P}$. The coalition number $C(G)$ is the maximum cardinality of a coalition partition of $G$. To represent a coalition partition $\mathcal{P}$ of $G$, a coalition graph $\CG(G, \mathcal{P})$ is created, where each vertex of the graph corresponds to a member of $\mathcal{P}$ and two vertices are adjacent if and only if their corresponding sets form a coalition in $G$. A coalition partition $\mathcal{P}$ of $G$ is a singleton coalition partition if every set in $\mathcal{P}$ consists of a single vertex. If a graph $G$ has a singleton coalition partition, then $G$ is referred to as a singleton-partition graph. A graph $H$ is called a singleton coalition graph of a graph $G$ if there exists a singleton coalition partition $\mathcal{P}$ of $G$ such that the coalition graph $\CG(G,\mathcal{P})$ is isomorphic to $H$. A singleton coalition graph chain with an initial graph $G_1$ is defined as the sequence $G_1\rightarrow G_2\rightarrow G_3\rightarrow\cdots$ where all graphs $G_i$ are singleton-partition graphs, and $\CG(G_i,\Gamma_1)=G_{i+1}$, where $\Gamma_1$ represents a singleton coalition partition of $G_i$. In this paper, we address two open problems posed by Haynes et al. We characterize all graphs $G$ of order $n$ and minimum degree $\delta(G)=2$ such that $C(G)=n$ and investigate the singleton coalition graph chain starting with graphs $G$ where $\delta(G)\le 2$.

The connected coalition in a graph $G=(V,E)$ consists of two disjoint sets of vertices $V_{1}$ and $V_{2}$, neither of which is a connected dominating set but whose union $V_{1}\cup V_{2}$, is a connected dominating set. A connected coalition partition in a graph $G$ of order $n=|V|$ is a vertex partition $\psi$ = $\{V_1, V_2,..., V_k \}$ such that every set $V_i \in \psi$ either is a connected dominating set consisting of a single vertex of degree $n-1$, or is not a connected dominating set but forms a connected coalition with another set $V_j\in \psi$ which is not a connected dominating set. The connected coalition number, denoted by $CC(G)$, is the maximum cardinality of a connected coalition partition of $G$. In this paper, we initiate the study of connected coalition in graphs and present some basic results. Precisely, we characterize all graphs that have a connected coalition partition. Moreover, we show that for any graph $G$ of order $n$ with $\delta(G)=1$ and with no full vertex, it holds that $CC(G)

Let $G$ be a graph with vertex set $V$. Two disjoint sets $V_1, V_2\subseteq V$ are called a total coalition in $G$, if neither $V_1$ and $V_2$ is a total dominating set of $G$ but $V_1\cup V_2$ is a total dominating set. A total coalition partition of $G$ is a vertex partition $\pi=\{V_1,V_2,\ldots, V_k\}$ such that no set of $\pi$ is a total dominating set but each set $V_i\in \pi$ forms a total coalition with another set $V_j\in \pi$. The maximum cardinality of a total coalition partition of $G$ is called the total coalition number of $G$, denoted by $TC(G)$. In this paper, we initiate the study of the total coalition in graphs and its properties., Comment: 12 pages

Let $G$ be a graph with vertex set $V$ and of order $n = |V|$, and let $\delta(G)$ and $\Delta(G)$ be the minimum and maximum degree of $G$, respectively. Two disjoint sets $V_1, V_2 \subseteq V$ form a coalition in $G$ if none of them is a dominating set of $G$ but their union $V_1\cup V_2$ is. A vertex partition $\Psi=\{V_1,\ldots, V_k\}$ of $V$ is a coalition partition of $G$ if every set $V_i\in \Psi$ is either a dominating set of $G$ with the cardinality $|V_i|=1$, or is not a dominating set but for some $V_j\in \Psi$, $V_i$ and $V_j$ form a coalition. The maximum cardinality of a coalition partition of $G$ is the coalition number $\mathcal{C}(G)$ of $G$. Given a coalition partition $\Psi = \{V_1, \ldots, V_k\}$ of $G$, a coalition graph $\CG(G, \Psi)$ is associated on $\Psi$ such that there is a one-to-one correspondence between its vertices and the members of $\Psi$, where two vertices of $\CG(G, \Psi)$ are adjacent if and only if the corresponding sets form a coalition in $G$. In this paper, we partially solve one of the open problems posed in Haynes et al. \cite{coal0} and we solve two open problems posed by Haynes et al. \cite{coal1}. We characterize all graphs $G$ with $\delta(G) \le 1$ and $\mathcal{C}(G)=n$, and we characterize all trees $T$ with $\mathcal{C}(T)=n-1$. We determine the number of coalition graphs that can be defined by all coalition partitions of a given path. Furthermore, we show that there is no universal coalition path, a path whose coalition partitions defines all possible coalition graphs.

Let $G=(V, E)$ be a simple and undirected graph. For some integer $k\geq 1$, a set $D\subseteq V$ is said to be a k-dominating set in $G$ if every vertex $v$ of $G$ outside $D$ has at least $k$ neighbors in $D$. Furthermore, for some real number $\alpha$ with $0<\alpha\leq1$, a set $D\subseteq V$ is called an $\alpha$-dominating set in $G$ if every vertex $v$ of $G$ outside $D$ has at least $\alpha\times d_v$ neighbors in $D$, where $d_v$ is the degree of $v$ in $G$. The cardinality of a minimum $k$-dominating set and a minimum $\alpha$-dominating set in $G$ is said to be the $k$-domination number and the $\alpha$-domination number of $G$, respectively. In this paper, we present some approximability and inapproximability results on the problem of finding $k$-domination number and $\alpha$-domination number of some classes of graphs. Moreover, we introduce a generalization of $\alpha$-dominating set which we call an $f$-dominating set. Given a function $f:\mathbb{N}\rightarrow \mathbb{R}$, where $\mathbb{N}=\{1, 2, 3, \ldots\}$, a set $D\subseteq V$ is said to be an $f$-dominating set in $G$ if every vertex $v$ of $G$ outside $D$ has at least $f(d_v)$ neighbors in $D$. We prove NP-hardness of the problem of finding a minimum $f$-dominating set in $G$, for a large family of functions $f$.