Fold thickness of some classes of graphs

Abstract

A 1-fold of $G$ is the graph $G'$ obtained from a graph $G$ by identifying two nonadjacent vertices in $G$ having at least one common neighbor and reducing the resulting multiple edges to simple edges. A sequence of graphs $G = G_0, G_1, G_2, \ldots ,G_k$, where $G_{i+1}$ is a 1-fold of $G_{i}$ for $i=0,1,2,\ldots ,k-1$ is called a uniform $k$-folding if all the graphs in the sequence are singular or all of them are nonsingular. The largest $k$ for which there exists a uniform $k$- folding of $G$ is called fold thickness of $G$ and it was first introduced in [Campe{\~n}a, F. J. H.; Gervacio, S. V. On the fold thickness of graphs. \emph{Arab, J. Math. (Springer)} {\bf9} (2020), no. 2, 345--355]. In this paper, we determine fold thickness of $K_n \odot \overline{K_m}$, $K_n + \overline{K_m}$, cone graph and tadpole graph.