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Planar wheel-like bricks

Fuliang Lu, Jinxin Xue

Abstract

An edge e in a matching covered graph G is removable if G-e is matching covered; a pair {e; f} of edges of G is a removable doubleton if G-e-f is matching covered, but neither G-e nor G-f is. Removable edges and removable doubletons are called removable classes, which was introduced by Lovasz and Plummer in connection with ear decompositions of matching covered graphs. A brick is a nonbipartite matching covered graph without nontrivial tight cuts. A brick G is wheel-like if G has a vertex h, such that every removable class of G has an edge incident with h. Lucchesi and Murty conjectured that every planar wheel-like brick is an odd wheel. We present a proof of this conjecture in this paper.

Planar wheel-like bricks

Abstract

An edge e in a matching covered graph G is removable if G-e is matching covered; a pair {e; f} of edges of G is a removable doubleton if G-e-f is matching covered, but neither G-e nor G-f is. Removable edges and removable doubletons are called removable classes, which was introduced by Lovasz and Plummer in connection with ear decompositions of matching covered graphs. A brick is a nonbipartite matching covered graph without nontrivial tight cuts. A brick G is wheel-like if G has a vertex h, such that every removable class of G has an edge incident with h. Lucchesi and Murty conjectured that every planar wheel-like brick is an odd wheel. We present a proof of this conjecture in this paper.

Paper Structure

This paper contains 4 sections, 27 theorems, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Lemmas
  4. Proof of Theorem \ref{['main']}

Key Result

Theorem 1.1

Every brick has at least $\Delta(G)$ removable classes, where $\Delta(G)$ is the maximum of the degrees of vertices in $G$.

Figures (3)

  • Figure 1: $\overline{C_6}$ (left) and $R_8$ (right).
  • Figure 2: Planar bricks with six vertices, where the bold edges are removable.
  • Figure 3: Illustration for the proof of Lemma \ref{['Wi_Wj']}, where the bold edges are removable.

Theorems & Definitions (43)

  • Theorem 1.1: CLM02c
  • Conjecture 1.2: Lucchesi2024
  • Theorem 1.3
  • Theorem 2.1: Tutte
  • Lemma 2.2: CLM12
  • Proposition 2.3
  • Theorem 2.4: Theorem 9.17 in Lucchesi2024
  • Theorem 2.5: CLM02II
  • Theorem 2.6: CLM02II
  • Proposition 2.7: CLM06
  • ...and 33 more