Surface embedding of non-bipartite $k$-extendable graphs

TL;DR

This paper determines the minimum extendability level for non-bipartite graphs embedded on surfaces, constructs specific examples like bow-tie graphs, and confirms the existence of infinitely many 3-extendable non-bipartite graphs on the Klein bottle.

Contribution

It introduces the surface extendability number for non-bipartite graphs on surfaces and constructs explicit examples, including bow-tie graphs, demonstrating their properties.

Findings

Minimum extendability number for non-bipartite surface-embeddable graphs identified

Constructed bow-tie graphs that are 3-extendable

Proved existence of infinitely many 3-extendable non-bipartite graphs on the Klein bottle

Abstract

We find the minimum number $k=\mu'(\Sigma)$ for any surface $\Sigma$, such that every $\Sigma$-embeddable non-bipartite graph is not $k$-extendable. In particular, we construct the so-called bow-tie graphs $C_6\bowtie P_n$, and show that they are $3$-extendable. This confirms the existence of an infinite number of $3$-extendable non-bipartite graphs which can be embedded in the Klein bottle.