On Coalition Graphs and Coalition Count of Graphs

TL;DR

This paper advances the study of coalition graphs and coalition count by providing a constructive method to realize any graph $G$ as a coalition graph $CG(H^*,\pi^*)\cong G$ using a compact base-vertex construction, addressing the open question of minimizing the order and size of $H^*$ in the realization. It refines the understanding of coalition count $c(G)$ by showing it is not comparable to the coalition number $C(G)$, and by establishing structural and numerical bounds, including a sharp bound $c(G)\ge d(G)-f$ involving the domatic number $d(G)$ and full vertices $f$. The authors characterize graphs with $c(G)=1$ precisely as those with $\alpha(G)=n-f$, and provide the exact form $G\cong (K_f+pK_1)\cup qK_1$, along with a specialized analysis for graphs with a single full vertex and $\delta(G)=1$ where $c(G)=s-2$ and $CG(G,\pi)\cong K_1\cup K_{1,s-2}$. They also introduce SP-graphs (singleton-partition graphs) and establish lower bounds linking $c(G)$ to $C(G)$ and $\alpha(G)$, enriching the landscape of coalition-related graph parameters and their interdependencies.

Abstract

Let $G$ be graph with vertex set $V(G)$ and order $n$. A coalition in a graph $G$ consists of two disjoint sets of vertices $V_1$ and $V_2$, neither of which is a dominating set but whose union $V_1 \cup V_2$ is a dominating set. A coalition partition, abbreviated $c$-partition, in a graph $G$ is a vertex partition $π=\left\{V_1 , V_2,\dots, V_k\right\}$ such that every set $V_i$ of $π$ is either a singleton dominating set, or is not a dominating set but forms a coalition with another set $V_j$ in $π$. The sets $V_i$ and $V_j$ are coalition partners in $G$. The coalition number $C(G)$ equals the maximum order $k$ of a $c$-partition of $G$. For any graph $G$ with a $c$-partition $π=\left\{V_1,V_2,\dots,V_k\right\}$, the coalition graph $CG(G,π)$ of $G$ is a graph with vertex set $V_1,V_2,\dots, V_k$, corresponding one-to-one with the set $π$, and two vertices $V_i$ and $V_j$ are adjacent in $CG(G,π)$ if and only if the sets $V_i$ and $V_j$ are coalition partners in $π$. In [4], authors proved that for every graph $G$ there exist a graph $H$ and $c$-partition $π$ such that $CG(H,π)\cong G$, and raised the question: Does there exist a graph $H^*$ of smaller order $n^*$ and size $m^*$ with a $c$-partition $π^*$ such that $CG(H^*,π^*)\cong G$?. In this paper, we constructed a graph $H^*$ of small order and size and a $c$- partition $π^*$ such that $CG(H^*,π^*)\cong G$. Recently, Haynes et al.[5] defined the coalition count $c(G)$ of a graph $G$ as the maximum number of different coalition in any $c$-partition of $G$. We characterize all graphs $G$ with $c(G)=1$. Further, imposing some suitable conditions on coalition number, we study the properties of coalition count of graph.

Theorems & Definitions (13)

• Example 1.1
• Theorem 2.1
• proof
• Theorem 3.1
• proof
• Theorem 3.2
• proof
• Corollary 3.3
• Theorem 3.4
• proof