Abstract. A graph [Formula: see text] is common if the limit as [Formula: see text] of the minimum density of monochromatic labeled copies of [Formula: see text] in an edge coloring of [Formula: see text] with red and blue is attained by a sequence of quasirandom colorings. We apply an information-theoretic approach to show that certain graphs obtained from odd cycles and paths via gluing operations are common. In fact, for every pair [Formula: see text] of such graphs, there exists [Formula: see text] such that an appropriate linear combination of red copies of [Formula: see text] and blue copies of [Formula: see text] is minimized by a quasirandom coloring in which [Formula: see text] edges are red; such a pair [Formula: see text] is said to be [Formula: see text] -common. Our approach exploits a strengthening of the common graph property for odd cycles that was recently proved using Schur convexity. We also exhibit a [Formula: see text]-common pair [Formula: see text] such that [Formula: see text] is uncommon.

Abstract. For [Formula: see text], the [Formula: see text]- neighbor bootstrap process in a graph [Formula: see text] starts with a set of infected vertices and, in each time step, every vertex with at least [Formula: see text] infected neighbors becomes infected. The initial infection percolates if every vertex of [Formula: see text] is eventually infected. We exactly determine the minimum cardinality of a set that percolates for the 3-neighbor bootstrap process when [Formula: see text] is a three-dimensional grid with minimum side-length at least 11. We also characterize the integers [Formula: see text] and [Formula: see text] for which there is a set of cardinality [Formula: see text] that percolates for the 3-neighbor bootstrap process in the [Formula: see text] grid; this solves a problem raised by Benevides et al. [HAL Research Report 03161419v4, 2021].