Let $G$ be a finite group which is not a cyclic $p$-group, $p$ a prime number. We define an undirected simple graph $Delta(G)$ whose vertices are the proper subgroups of $G$, which are not contained in the Frattini subgroup of $G$ and two vertices $H$ and $K$ are joined by an edge if and only if $G=langle H , Krangle$. In this paper we classify finite groups with planar graph. %For this, by Kuratowski's Theorem, we have to study subdivisions %of the Kuratowski graphs $K_{3 , 3}$ and $K_5$ in the graph $Delta(G)$. Our result shows that only few groups have planar graphs.
Taeri, B., & Ahmadi, H. (2014). On the planarity of a graph related to the join of subgroups of a finite group. Bulletin of the Iranian Mathematical Society, 40(6), 1413-1431.
MLA
Taeri, B., & Ahmadi, H. "On the planarity of a graph related to the join of subgroups of a finite group", Bulletin of the Iranian Mathematical Society, 40, 6, 2014, 1413-1431.
HARVARD
Taeri B., Ahmadi H. (2014). 'On the planarity of a graph related to the join of subgroups of a finite group', Bulletin of the Iranian Mathematical Society, 40(6), pp. 1413-1431.
CHICAGO
B. Taeri & H. Ahmadi, "On the planarity of a graph related to the join of subgroups of a finite group," Bulletin of the Iranian Mathematical Society, 40 6 (2014): 1413-1431,
VANCOUVER
Taeri B., Ahmadi H. On the planarity of a graph related to the join of subgroups of a finite group. BIMS. 2014;40(6):1413-1431.
