Chasing robbers on random graphs: Zigzag theorem

Abstract

ABSTRACT:In this paper, we study the vertex pursuit game of Cops and Robbers where cops try to capture a robber on the vertices of the graph. The minimum number of cops required to win on a given graph G is the cop number of G. We present asymptotic results for the game of Cops and Robber played on a random graph G(n, p) for a wide range of p = p(n). It has been shown that the cop number as a function of an average degree forms an intriguing zigzag shape.

Meyniel's conjecture holds for random graphs

“…c(G(n, p)) = n 1/2+o (1) . Somewhat surprisingly, between these values c(G(n, p)) was shown by Luczak and the first author [15] to have more complicated behaviour. Theorem 1.1 ( [15]).…”

“…Somewhat surprisingly, between these values c(G(n, p)) was shown by Luczak and the first author [15] to have more complicated behaviour. Theorem 1.1 ( [15]). Let 0 < α < 1, let j ≥ 1 be integer, and let d = d(n) = (n − 1)p = n α+o (1) .…”

“…Let r = r(d) be the smallest integer such that d r+1 ≥ √ n. Note that it follows from d r < √ n, and assumption (i) that |N (v, r)| < 2 √ n. We consider two cases, depending on the value of d r+1 , and in each case we give a strategy which permits the cops to win a.a.s. The first case is based on the idea used in [15]. Case 1. d r+1 ≥ √ n log n. This case is rather easy.…”

“…First of all, note that only the special values of p in Case 2 in the proof of Theorem 3.1 required much care, and it is reasonable to suppose that a bit more work, perhaps combining our approach with that in [15], would easily produce a sharper result in Theorem 1.1(ii), namely, that the upper bound can be made a constant times the lower bound. However, in this paper we restrict ourselves to our main purpose of showing that Meyniel's conjecture holds a.a.s.…”

Cops and Robbers playing on edges

“…if p tends to one as n goes to infinity (see [25] for this and stronger results). Somewhat surprisingly, it was shown by Luczak and the third author [22] that for sparse random graphs the cop number has more complicated behaviour. It follows that a.a.s.…”