The bondage number of graphs on topological surfaces and Teschner's conjecture

TL;DR

This paper establishes new bounds on the bondage number of graphs embedded on topological surfaces, advancing understanding of Teschner's conjecture and providing tight bounds related to surface genus and graph properties.

Contribution

It offers improved upper bounds for the bondage number based on surface genus and vertex degree, and confirms Teschner's conjecture for most graphs.

Findings

Constant upper bounds for graphs on surfaces.

Tight lower bounds for vertex counts on surfaces.

Confirmation of Teschner's conjecture for almost all graphs.

Abstract

The bondage number of a graph is the smallest number of its edges whose removal results in a graph having a larger domination number. We provide constant upper bounds for the bondage number of graphs on topological surfaces, improve upper bounds for the bondage number in terms of the maximum vertex degree and the orientable and non-orientable genera of the graph, and show tight lower bounds for the number of vertices of graphs 2-cell embeddable on topological surfaces of a given genus. Also, we provide stronger upper bounds for graphs with no triangles and graphs with the number of vertices larger than a certain threshold in terms of the graph genera. This settles Teschner's Conjecture in positive for almost all graphs.