Characterization of Joined Graphs

Abstract

The join of simple graphs $G_1$ and $G_2$, written by $G_1\vee G_2$, is the graph obtained from the disjoint union between $G_1$ and $G_2$ by adding the edges $\{xy:x\in V(G_1),y\in V(G_2)\}$. We call a simple graph $G$ as a joined graph if there are $G_1$ and $G_2$ that $G=G_1\vee G_2$. In this paper, we give conditions to determine that which graphs are joined graphs and use its properties to investigate the chromatic number of joined graphs.