Planarity of the bipartite graph associated to squares of elements and subgroups of a group

A new kind of bipartite graph is defined in this paper. The work is motivated mainly from a graph introduced by Al-Kaseasbeh and Erfanian [A bipartite graph associated to elements and cosets of subgroups of a finite group, AIMS Math. 6(10) (2021) 10395–10404] and partially from the undirected power graph of a group and from the inclusion graph. For a nontrivial group $G$, we define a simple undirected bipartite graph $\mathrm{\Gamma }(G)$ with the vertex set $V(\mathrm{\Gamma }(G))=A\cup B$, where $A$ is the set of all elements of the group $G$ and $B$ is the set of all subgroups $H$ of $G$ such that $H\ne \{e\}$ and two vertices $a\in A$ and $H\in B$ are adjacent if and only if $a^2\in H$. Here, we classify all the finite groups $G$ whose bipartite graphs $\mathrm{\Gamma }(G)$ are planar. In addition, we also classify the finite groups $G$ such that $\mathrm{\Gamma }(G)$ is outerplanar, maximal planar, maximal outerplanar, star, tree, claw graph, respectively. The planarity of the graph $\mathrm{\Gamma }(G)$ corresponding to an infinite group $G$ has also been investigated here. It is also observed that $\mathrm{\Gamma }(G)$ satisfies Vizing’s conjecture.