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Totally odd immersions in line graphs

Andrea Jiménez, Daniel A. Quiroz, Christopher Thraves Caro

Abstract

The immersion-analogue of Hadwiger's Conjecture states that every graph $G$ contains an immersion of $K_{χ(G)}$. This conjecture has been recently strengthened in the following way: every graph $G$ contains a totally odd immersion of $K_{χ(G)}$. We prove this stronger conjecture for line graphs of constant-multiplicity multigraphs, thus extending a result of Guyer and McDonald.

Totally odd immersions in line graphs

Abstract

The immersion-analogue of Hadwiger's Conjecture states that every graph contains an immersion of . This conjecture has been recently strengthened in the following way: every graph contains a totally odd immersion of . We prove this stronger conjecture for line graphs of constant-multiplicity multigraphs, thus extending a result of Guyer and McDonald.
Paper Structure (2 sections, 6 theorems)

This paper contains 2 sections, 6 theorems.

Table of Contents

  1. Introduction
  2. The proof

Key Result

Theorem 1

Every (simple) graph $G$ with maximum degree $d$ and chromatic index $d + 1$, contains two vertices $x,y$ and a collection of $d$ pairwise edge-disjoint paths between $x$ and $y$.

Theorems & Definitions (11)

  • Theorem 1: Thomassen T07
  • Conjecture 2: Churchley C17
  • Theorem 3
  • Conjecture 4
  • Lemma 5: Vizing
  • Theorem 6
  • proof
  • Claim 6.1
  • Lemma 7: Guyer and McDonald GMcD19
  • Theorem 8
  • ...and 1 more