Cops and robber in graphs with bounded vertex cover number

Meyniel’s conjecture states that $n$-vertex connected graphs have cop number $O(\sqrt{n})$. The current best known upper bound is $n/2^{(1-o(1))\sqrt{\log n}}$, proved independently by Lu and Peng (2011), and by Scott and Sudakov (2011). In this paper, we extend their result by showing that every connected graph with vertex cover number $k$ has cop number at most $k/2^{(1-o(1))\sqrt{\log k}}$. This is the first sublinear upper bound on the cop number in terms of the vertex cover number.

💡 Research Summary

The paper addresses the classic Cops and Robber pursuit–evasion game on graphs, focusing on the cop number cop(G), the minimum number of cops needed to guarantee capture of a robber on a connected graph G. While Meyniel’s conjecture (1985) predicts cop(G)=O(√n) for any n‑vertex connected graph, the best known general bound to date is n / 2^{(1‑o(1))√{log n}} (Lu–Peng and Scott–Sudakov, 2011). The authors ask whether a sublinear bound can be expressed in terms of a structural parameter that is typically much smaller than n. They answer this affirmatively by proving a sublinear bound in terms of the vertex‑cover number vc(G).

Main Results

1. Theorem 1: For every connected graph G,
   \