New upper bounds for the bondage number of a graph in terms of its maximum degree and euler characteristic

The bondage number 6(G) of a graph G is the smallest number of edges whose removal from G results in a graph with larger domination number. Let G be embeddable on a surface whose Euler characteristic χ is as large as possible, and assume χ < 0. Gagarin-Zverovich and Huang have recently found upper bounds of 6(G) in terms of the maximum degree Δ(G) and the Euler characteristic x-In this paper we prove a better upper bound 6(G) ≤ Δ(G) + [t] where t is the largest real root of the cubic equation z³ + z² + (3χ -8)z + 9x - 12 = 0; this upper bound is asymptotically equivalent to 6(G) ≤ A(G) + 1. We also establish further improved upper bounds for 6(G) when the girth, order, or size of the graph G is large compared with |χ|.