Forbidden subgraphs of power graphs

Pallabi Manna¹ ; Peter J. Cameron² ; Ranjit Mehatari³

¹Department of Mathematics, NIT Rourkela, Odisha-769008
²School of Mathematics and Statistics, University of St Andrews, North Haugh, St Andrews, Fife, KY16 9SS, UK
³Department of Mathematics, NIT Rourkela

The electronic journal of combinatorics, Tome 28 (2021) no. 3

Résumé

The undirected power graph (or simply power graph) of a group $G$, denoted by $P(G)$, is a graph whose vertices are the elements of the group $G$, in which two vertices $u$ and $v$ are connected by an edge between if and only if either $u=v^i$ or $v=u^j$ for some $i$, $j$. A number of important graph classes, including perfect graphs, cographs, chordal graphs, split graphs, and threshold graphs, can be defined either structurally or in terms of forbidden induced subgraphs. We examine each of these five classes and attempt to determine for which groups $G$ the power graph $P(G)$ lies in the class under consideration. We give complete results in the case of nilpotent groups, and partial results in greater generality. In particular, the power graph is always perfect; and we determine completely the groups whose power graph is a threshold or split graph (the answer is the same for both classes). We give a number of open problems.

Détail

DOI : 10.37236/9961

Classification : 05C25, 05C60, 05C17