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Skewness, crossing number and Euler's bound for graphs on surfaces

Paul C. Kainen

TL;DR

The paper analyzes the relationship between Euler-based lower bounds and graph invariants on orientable surfaces, focusing on the inequalities $\delta_S(G) \leq \mu_S(G) \leq \nu_S(G)$ and their integer variants. It surveys existing results and extends them with new findings for the folded cube, establishing broad equalities $\varepsilon_t(G)=\mu_t(G)$ for key families and providing constructive embeddings via triangulations and quadrangulations. The results cover complete graphs and complete bipartite graphs on sphere, torus, and higher-genus surfaces, as well as cubes and folded cubes, showing that the Euler-based bound is often tight or nearly so. Methodologically, the work combines genus calculations, explicit drawings, and quadrilateral decompositions (VDQC) to realize embeddings and obtain exact or near-exact equalities. The findings advance understanding of when the Euler lower bound coincides with skewness and crossing numbers, with implications for graph drawing on surfaces and potential Vizing-type phenomena in related invariants.

Abstract

For every connected graph $G$ and surface $S$, we consider the well-known string of inequalities $δ_S(G) \leq μ_S(G) \leq ν_S(G)$, where $μ$ and $ν$ denote skewness and crossing number and $δ$ is the Euler-formula lower bound. Recent developments are surveyed; new results are given for the ``folded'' cube including its genus.

Skewness, crossing number and Euler's bound for graphs on surfaces

TL;DR

The paper analyzes the relationship between Euler-based lower bounds and graph invariants on orientable surfaces, focusing on the inequalities and their integer variants. It surveys existing results and extends them with new findings for the folded cube, establishing broad equalities for key families and providing constructive embeddings via triangulations and quadrangulations. The results cover complete graphs and complete bipartite graphs on sphere, torus, and higher-genus surfaces, as well as cubes and folded cubes, showing that the Euler-based bound is often tight or nearly so. Methodologically, the work combines genus calculations, explicit drawings, and quadrilateral decompositions (VDQC) to realize embeddings and obtain exact or near-exact equalities. The findings advance understanding of when the Euler lower bound coincides with skewness and crossing numbers, with implications for graph drawing on surfaces and potential Vizing-type phenomena in related invariants.

Abstract

For every connected graph and surface , we consider the well-known string of inequalities , where and denote skewness and crossing number and is the Euler-formula lower bound. Recent developments are surveyed; new results are given for the ``folded'' cube including its genus.
Paper Structure (6 sections, 15 theorems, 16 equations)

This paper contains 6 sections, 15 theorems, 16 equations.

Table of Contents

  1. Introduction
  2. Definitions and lemmas
  3. $K_n$ and $K_{a,b}$ in the torus and the sphere
  4. $K_n$ and $K_{a,b}$ on higher genus surfaces
  5. Cubes and folded cubes
  6. Discussion

Key Result

Lemma 1

For all graphs $G$ and $t \geq 0$, $\delta_t(G) \leq \varepsilon_t(G) \leq \mu_t(G) \leq \nu_t(G)$.

Theorems & Definitions (25)

  • Lemma 1: pck-lowerbound
  • Lemma 2
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • Theorem 5
  • proof
  • Theorem 6
  • proof
  • ...and 15 more