General sharp upper bounds on the total coalition number

TL;DR

This paper establishes a sharp upper bound on the total coalition number of a graph based on its maximum degree, explores the structure of total coalition graphs, and demonstrates that any graph can be realized as such.

Contribution

The paper introduces a general sharp upper bound on the total coalition number related to maximum degree and studies the properties of total coalition graphs.

Findings

Derived a sharp upper bound on total coalition number

Analyzed the structure of total coalition graphs

Proved every graph can be realized as a total coalition graph

Abstract

Let $G(V,E)$ be a finite, simple, isolate-free graph. Two disjoint sets $A,B\subset V$ form a total coalition in $G$, if none of them is a total dominating set, but their union $A\cup B$ is a total dominating set. A vertex partition $\Psi=\{C_1,C_2,\dots,C_k\}$ is a total coalition partition, if none of the partition classes is a total dominating set, meanwhile for every $i\in\{1,2,\dots,k\}$ there exists a distinct $j\in\{1,2,\dots,k\}$ such that $C_i$ and $C_j$ form a total coalition. The maximum cardinality of a total coalition partition of $G$ is the total coalition number of $G$ and denoted by $TC(G)$. We give a general sharp upper bound on the total coalition number as a function of the maximum degree. We further investigate this optimal case and study the total coalition graph. We show that every graph can be realised as a total coalition graph.