On the fold thickness of graphs

Abstract

AbstractThe graph $$G'$$ G ′ obtained from a graph G by identifying two nonadjacent vertices in G having at least one common neighbor is called a 1-fold of G. A sequence $$G_0, G_1, G_2, \ldots , G_k$$ G 0 , G 1 , G 2 , … , G k of graphs such that $$G_0=G$$ G 0 = G and $$G_i$$ G i is a 1-fold of $$G_{i-1}$$ G i - 1 for each $$i=1, 2, \ldots , k$$ i = 1 , 2 , … , k is called a uniform k-folding of G if the graphs in the sequence are all singular or all nonsingular. The fold thickness of G is the largest k for which there is a uniform k-folding of G. We show here that the fold thickness of a singular bipartite graph of order n is $$n-3$$ n - 3 . Furthermore, the fold thickness of a nonsingular bipartite graph is 0, i.e., every 1-fold of a nonsingular bipartite graph is singular. We also determine the fold thickness of some well-known families of graphs such as cycles, fans and some wheels. Moreover, we investigate the fold thickness of graphs obtained by performing operations on these families of graphs. Specifically, we determine the fold thickness of graphs obtained from the cartesian product of two graphs and the fold thickness of a disconnected graph whose components are all isomorphic.