Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Decomposability of regular graphs to $4$ locally irregular subgraphs

Jakub Przybyło

Abstract

A locally irregular graph is a graph whose adjacent vertices have distinct degrees. It was conjectured that every connected graph is edge decomposable to $3$ locally irregular subgraphs, unless it belongs to a certain family of exceptions, including graphs of small maximum degrees, which are not decomposable to any number of such subgraphs. Recently Sedlar and Škrekovski exhibited a counterexample to the conjecture, which necessitates a decomposition to (at least) $4$ locally irregular subgraphs. We prove that every $d$-regular graph with $d$ large enough, i.e. $d\geq 54000$, is decomposable to $4$ locally irregular subgraphs. Our proof relies on a mixture of a numerically optimized application of the probabilistic method and certain deterministic results on degree constrained subgraphs due to Addario-Berry, Dalal, McDiarmid, Reed, and Thomason, and to Alon and Wei, introduced in the context of related problems concerning irregular subgraphs.

Decomposability of regular graphs to $4$ locally irregular subgraphs

Abstract

A locally irregular graph is a graph whose adjacent vertices have distinct degrees. It was conjectured that every connected graph is edge decomposable to locally irregular subgraphs, unless it belongs to a certain family of exceptions, including graphs of small maximum degrees, which are not decomposable to any number of such subgraphs. Recently Sedlar and Škrekovski exhibited a counterexample to the conjecture, which necessitates a decomposition to (at least) locally irregular subgraphs. We prove that every -regular graph with large enough, i.e. , is decomposable to locally irregular subgraphs. Our proof relies on a mixture of a numerically optimized application of the probabilistic method and certain deterministic results on degree constrained subgraphs due to Addario-Berry, Dalal, McDiarmid, Reed, and Thomason, and to Alon and Wei, introduced in the context of related problems concerning irregular subgraphs.
Paper Structure (10 sections, 7 theorems, 62 equations)

This paper contains 10 sections, 7 theorems, 62 equations.

Table of Contents

  1. Introduction
  2. Outline of the main ideas
  3. Tools
  4. Proof of Theorem \ref{['MainResult-4-decomposability_of_regular']}
  5. Random colour assignment and distinguished sets
  6. Assuring desired features of the distinguished sets
  7. Decomposition of $G$ to two similar subgraphs
  8. A decomposition of $H_0$ to two locally irregular subgraphs
  9. Justification of no degree conflicts in $F_1$ and $F_2$
  10. Final remarks

Key Result

Theorem 2

Each $d$-regular graph with $d\geq 10^7$ is decomposable to $3$ locally irregular subgraphs.

Theorems & Definitions (11)

  • Conjecture 1: LocalIrreg_1
  • Theorem 2: LocalIrreg_1
  • Conjecture 3: SedlarSkrekovski-LocIrreg
  • Theorem 4
  • Theorem 5: Chernoff Bound
  • Theorem 6: McDiarmid's Inequality
  • Theorem 7: Lovász Local Lemma
  • Lemma 8
  • Lemma 9
  • Claim 10
  • ...and 1 more