On Kainen's conjectures on surface crossing numbers

TL;DR

This paper investigates Kainen's conjectures on the crossing numbers of graphs on surfaces, confirming their validity for most cases and identifying specific exceptions, thus advancing understanding of graph embeddings in topological surfaces.

Contribution

It proves Kainen's conjectures are true for most graphs and surfaces, with specific exceptions, and extends results to nonorientable surfaces.

Findings

Kainen's conjectures hold for all but three specific graphs.

Confirmed nonorientable analogues with two exceptions.

Provides bounds and exact crossing numbers for various graphs.

Abstract

In 1972, Kainen proved a general lower bound on the crossing number of a graph in a closed surface and conjectured that this bound is tight when the graph is either a complete graph or a complete bipartite graph, and the surface is of genus close to the minimum genus of that graph. Prior to the present work, these conjectures were known to be true only for small cases and when the conjectures predict a crossing number of 0, i.e., when a triangular or quadrangular embedding was already known. We show that Kainen's conjectures are true except for the three graphs $K_9$, $K_{3,5}$, and $K_{5,5}$. We also prove nonorientable analogues of these conjectures, where the only exceptions to the general formulas are $K_7$ and $K_8$.