The maximum and minimum genus of a multibranched surface

TL;DR

This paper establishes bounds for the maximum and minimum genus of multibranched surfaces using Betti numbers and boundary genus, linking graph theory with 3-manifold topology.

Contribution

It introduces new bounds for the genus of multibranched surfaces and relates these to properties of graph products with S^1, bridging graph theory and 3-manifold theory.

Findings

Maximum and minimum genus bounds based on Betti number and boundary genus

Maximum and minimum genus of G×S^1 relate to those of G, doubled

Provides a new connection between graph theory and 3-manifold topology

Abstract

In this paper, we give a lower bound for the maximum and minimum genus of a multibranched surface by the first Betti number and the minimum and maximum genus of the boundary of the neighborhood of it, respectively. As its application, we show that the maximum and minimum genus of $G\times S^1$ is equal to twice of the maximum and minimum genus of $G$ for a graph $G$, respectively. This provides an interplay between graph theory and 3-manifold theory.