Embeddings of four-valent framed graphs into two-surfaces

TL;DR

This paper investigates the minimal and maximal genus surfaces into which four-valent graphs with fixed opposite edge structures can be embedded, offering new combinatorial and knot theoretic reformulations.

Contribution

It introduces new relations and reformulations for embedding four-valent graphs into surfaces, advancing understanding in combinatorics and knot theory.

Findings

Derived partial relations for graph embeddings

Reformulated embedding problems in combinatorial language

Connected graph embedding problems to knot theory concepts

Abstract

We study the problem of finding the minimal (maximal) genus for a surface where a given four-valent graph with fixed opposite edge structure can be embedded into. We find several partial relations and give new reformulations in combinatorial and knot theoretic languages.