Zero-forcing in random regular graphs

Abstract

The zero forcing process is an iterative graph colouring process in which at each time step a coloured vertex with a single uncoloured neighbour can force this neighbour to become coloured. A zero forcing set of a graph is an initial set of coloured vertices that can eventually force the entire graph to be coloured. The zero forcing number is the size of the smallest zero forcing set. We explore the zero forcing number for random regular graphs, improving on bounds given by Kalinowski, Kamcev and Sudakov [15]. We also propose and analyze a degree greedy algorithm for finding small zero forcing sets using the differential equations method.

Tight Bounds on Probabilistic Zero Forcing on Hypercubes and Grids

“…Of course, Theorem 2 can be improved immediately via (1) and the fact that for 0 < p < 1 constant, the radius of G(n, p) is a.a.s. 2 (see e.g.…”

“…In [9], the authors explored probabilistic zero forcing on G(n, p) in more detail and, in particular, proved the above conjecture. Their results can be summarized in the following theorem that shows that probabilistic zero forcing occurs much faster in G(n, p) than in a general graph G, as evidenced by the bounds of (1).…”

“…Since the state of D j+1 at time t j+1 (namely, configuration C j+1 ) depends only on the state of D j at time t j (namely, configuration C j ) independent of the time and the choice of vertices, we can capture the behaviour of the process using a Markov chain (C j ) ∞ j=0 on state space Ω = {0, 1} d . For example, when d = 2 the Markov chain has transition matrix where the states are in the order (0, 0), (0, 1), (1, 0), (1,1). It is easy to see that for any d this Markov chain is irreducible and aperiodic, and so it has a limiting distribution which is equal to the stationary distribution.…”

“…Zero forcing has sparked a lot of interest recently. Some work has been done on calculating or bounding the zero forcing number for specific structures such as graph products [12], graphs with large girth [8] and random graphs [1,16], while others have studied variants of zero forcing such as connected zero forcing [4] or positive semi-definite zero forcing [2].…”