The matching book embedding of the F-sum of two graphs

The $F$-sum is a new graph operation defined by combining four graph transformation operations with the Cartesian product operation. A matching book embedding of a graph $G$ is a book embedding in which the vertices of $G$ are placed on a fixed linear order along the spine, and the edges are assigned to pages such that (i) no two edges on the same page cross, and (ii) each vertex has degree at most one on every page. The minimum number of pages required for such a matching book embedding is called the \emph{matching book thickness} of $G$, denoted by $mbt(G)$. A graph $G $ is dispersable if and only if $ mbt(G) = \Delta(G) $, and nearly dispersable if and only if $mbt(G) = \Delta(G) + 1 $. In this paper, we determine the dispersability of outerplanar graphs and establish an upper bound on the matching book thickness of the $F$-sum of any simple graph with any dispersable bipartite graph.