The average Euler-genus of the vertex-amalgamation of signed graphs

TL;DR

This paper extends existing theorems on face counting and genus bounds from graphs to signed graphs, providing new insights into their embeddings and topological properties.

Contribution

It generalizes face counting theorems to all surfaces and extends genus bounds to signed graphs, broadening the understanding of graph embeddings.

Findings

Generalized face counting theorem to all surfaces.

Extended genus bounds to signed graphs.

Provided new formulas for average Euler-genus.

Abstract

In this paper, we first generalize a theorem for counting the number of faces of an oriented embedding of a graph that passing through a given cut-edge set [S. Stahl, Trans. Amer. Math. Soc. 259 (1980), 129--145] to all surfaces. Then we extend Stahl's bounds for the average genus of the vertex-amalgamation of graphs [S. Stahl, Discrete Math. 142 (1995), 235--245] to signed graphs.