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Some conjectures on $r$-graphs and equivalences

Yulai Ma, Eckhard Steffen, Isaak H. Wolf, Junxue Zhang

Abstract

An $r$-regular graph is an $r$-graph, if every odd set of vertices is connected to its complement by at least $r$ edges. Seymour [On multicolourings of cubic graphs, and conjectures of Fulkerson and Tutte.~\emph{Proc.~London Math.~Soc.}~(3), 38(3): 423-460, 1979] conjectured (1) that every planar $r$-graph is $r$-edge colorable and (2) that every $r$-graph has $2r$ perfect matchings such that every edge is contained in precisely two of them. We study several variants of these conjectures. A $(t,r)$-PM is a multiset of $t \cdot r$ perfect matchings of an $r$-graph $G$ such that every edge is in precisely $t$ of them. We show that the following statements are equivalent for every $t, r \geq 1$: 1. Every planar $r$-graph has a $(t,r)$-PM. 2. Every $K_5$-minor-free $r$-graph has a $(t,r)$-PM. 3. Every $K_{3,3}$-minor-free $r$-graph has a $(t,r)$-PM. 4. Every $r$-graph whose underlying simple graph has crossing number at most $1$ has a $(t,r)$-PM.

Some conjectures on $r$-graphs and equivalences

Abstract

An -regular graph is an -graph, if every odd set of vertices is connected to its complement by at least edges. Seymour [On multicolourings of cubic graphs, and conjectures of Fulkerson and Tutte.~\emph{Proc.~London Math.~Soc.}~(3), 38(3): 423-460, 1979] conjectured (1) that every planar -graph is -edge colorable and (2) that every -graph has perfect matchings such that every edge is contained in precisely two of them. We study several variants of these conjectures. A -PM is a multiset of perfect matchings of an -graph such that every edge is in precisely of them. We show that the following statements are equivalent for every : 1. Every planar -graph has a -PM. 2. Every -minor-free -graph has a -PM. 3. Every -minor-free -graph has a -PM. 4. Every -graph whose underlying simple graph has crossing number at most has a -PM.

Paper Structure

This paper contains 6 sections, 13 theorems, 2 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Basic results
  3. Clique-sums
  4. $K_5$-minor-free or $K_{3,3}$-minor-free graphs
  5. $r$-graphs
  6. Proof of Theorem \ref{['Theorem-equiv-minor-free']}

Key Result

Theorem 1.1

Let $r \geq 3$. The following statements are equivalent.

Figures (1)

  • Figure 1: The Wagner graph $V_8$

Theorems & Definitions (25)

  • Theorem 1.1: jaeger1980tait
  • Conjecture 1.2: seymour1979multi
  • Conjecture 1.3
  • Theorem 1.4: ellingham1984petersen
  • Conjecture 1.5: seymour1979multi
  • Conjecture 1.6
  • Conjecture 1.7
  • Conjecture 1.8
  • Conjecture 1.9
  • Theorem 1.10
  • ...and 15 more