The Kuratowski covering conjecture for graphs of order < 10 for the nonorientable surfaces of genus 3 and 4

TL;DR

This paper verifies the Kuratowski covering conjecture for all minimal forbidden subgraphs of order less than 10 on nonorientable surfaces of genus 3 and 4, extending understanding of graph embeddings.

Contribution

It confirms the conjecture for small forbidden subgraphs on nonorientable surfaces of genus 3 and 4, providing new validation for the conjecture in these cases.

Findings

Conjecture holds for all minimal forbidden subgraphs of order < 10 for genus 3 and 4.

Identifies specific structures of forbidden subgraphs in these cases.

Supports the broader applicability of the Kuratowski covering conjecture.

Abstract

Kuratowski proved that a finite graph embeds in the plane if it does not contain a subdivision of either K_5 or K_{3,3}, called Kuratowski subgraphs. A conjectured generalization of this result to all nonorientable surfaces says that a finite minimal forbidden subgraph for the nonorientable surface of genus g can be written as the union of g+1 Kuratowski subgraphs such that the union of each pair of these fails to embed in the projective plane, the union of each triple of these fails to embed in the Klein bottle if g >= 2, and the union of each triple of these fails to embed in the torus if g >= 3. We show that this conjecture is true for all minimal forbidden subgraphs of order < 10 for the nonorientable surfaces of genus 3 and 4.