If is a graph and , we let denote a subgraph of where each edge of appears in with independently with probability .
Say that a family of graphs is -bounded if there exists a function so that every satisfies .
Keywords: polyhedral graphs, distribution
P vs. PSPACE ★★★
Given is a 3SAT (3CNF) formula on variables, for some , and clauses drawn uniformly at random from the set of formulas on variables. Return with probability at least 0.5 (over the instances) that is typical without returning typical for any instance with at least simultaneously satisfiable clauses.
Let be a finite undirected simple graph.
A -page book embedding of consists of a linear order of and a (non-proper) -colouring of such that edges with the same colour do not cross with respect to . That is, if for some edges , then and receive distinct colours.
One can think that the vertices are placed along the spine of a book, and the edges are drawn without crossings on the pages of the book.
The book thickness of , denoted by bt is the minimum integer for which there is a -page book embedding of .
Let be the graph obtained by subdividing each edge of exactly once.
It would follow from the following stronger conjecture [Da]: