Meyniel's conjecture holds for random graphs

Abstract

Abstract. In the game of cops and robber, the cops try to capture a robber moving on the vertices of the graph. The minimum number of cops required to win on a given graph G is called the cop number of G. The biggest open conjecture in this area is the one of Meyniel, which asserts that for some absolute constant C, the cop number of every connected graph G is at most C |V (G)|. In this paper, we show that Meyniel's conjecture holds asymptotically almost surely for the binomial random graph G(n, p), which improves upon existing results showing that asymptotically almost surely the cop number of G(n, p) is O( √ n log n) provided that pn ≥ (2 + ε) log n for some ε > 0. We do this by first showing that the conjecture holds for a general class of graphs with some specific expansion-type properties. This will also be used in a separate paper on random d-regular graphs, where we show that the conjecture holds asymptotically almost surely when d = d(n) ≥ 3.

The Game of Overprescribed Cops and Robbers Played on Graphs

“…Hence, only up to this point the lower bound for the number of cops is a decreasing function of d. After removing this artificial restriction on d, there would be more choices for d to satisfy the desired condition but clearly one should consider the smallest value of d to get the best bound. We observe that since Q n is a relatively good expander, the proof follows similar ideas as the ones used to bound the cop number for random graphs [13,19,20].…”

“…Using ideas from [13,20], we may obtain bounds for the capture time of binomial random graphs. For simplicity, we restrict ourselves to dense random graphs (d = p(n − 1) ≥ log 3 n) and a large number of cops (k = k(n) ≥ C √ n log n).…”

“…Further, we present a sketch of the proof only. Adjusting the argument to sparser graphs (based on the more sophisticated argument in [20] for the sparse case) should be straightforward, but the upper bound will not match the lower bound. Similarly, adjusting the argument to fewer cops is possible but definitely not all the way to the cop number.…”

Chasing a Fast Robber on Planar Graphs and Random Graphs

“…The original Cops and Robber game for Erdős–Rényi random graphs has been studied by several authors . In particular, Prałat and Wormald proved that Meyniel's conjecture holds for random graphs. In this section, we prove Theorem 2 that determines the typical asymptotic behavior of for the Erdős–Rényi random graph.…”

Cops and Robbers on diameter two graphs

“…This conjecture has been almost completely established for random graphs by Bollobás, Kun, Leader [2], Luczak, Pra lat [11] and Pra lat, Wormald [14]. For general graphs, even…”