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Circuits through prescribed edges

Paul Knappe, Max Pitz

Abstract

We prove that a connected graph contains a circuit---a closed walk that repeats no edges---through any $k$ prescribed edges if and only if it contains no odd cut of size at most $k$.

Circuits through prescribed edges

Abstract

We prove that a connected graph contains a circuit---a closed walk that repeats no edges---through any prescribed edges if and only if it contains no odd cut of size at most .

Paper Structure

This paper contains 5 sections, 13 theorems, 4 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A reduction to the bridge case
  4. Proving the bridge case
  5. Concluding remarks and an open question

Key Result

Theorem 1.2

For any set $S$ of $k$ independent edges in a $(k+1)$-connected graph, there is a cycle in $G$ containing $S$.

Figures (3)

  • Figure 1: A ladder with specified rungs $S=\{e_1,\dots,e_k\}$.
  • Figure 2: A circuit $H=H_1e_1H_2e_2H_3e_3$ with segments $H_1,H_2,H_3$.
  • Figure 3: Obtaining the rerouted trail $Q'$ from $Q$.

Theorems & Definitions (40)

  • Conjecture 1.1: Lovász-Woodall Conjecture
  • Theorem 1.2: Häggkvist and Thomassen
  • Theorem 1.3: Kawarabayashi
  • Theorem 1.4
  • Corollary 1.5
  • Theorem 1.6: Jaeger
  • Example 1.7: Counterexample to MR1812342*Theorem 1.1 & 4.1
  • proof
  • Theorem 1.8: Lai
  • Definition 2.1
  • ...and 30 more