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Progress towards generalized Nash-Williams' conjecture on $K_4$-decompositions

Menglong Zhang, Gennian Ge

Abstract

A $K_4$-decomposition of a graph is a partition of its edges into $K_4$s. A fractional $K_4$-decomposition is an assignment of a nonnegative weight to each $K_4$ in a graph such that the sum of the weights of the $K_4$s containing any given edge is one. Formulating a nonlinear programming and reducing the number of variables slowly, we prove that every graph on $n$ vertices with minimum degree at least $\frac{31}{33}n$ has a fractional $K_4$-decomposition. This improves a result of Montgomery that the same conclusion holds for graphs with minimum degree at least $\frac{399}{400}n$. Together with a result of Barber, Kühn, Lo, and Osthus, this result implies that for all $\varepsilon> 0$, every large enough $K_4$-divisible graph on $n$ vertices with minimum degree at least $(\frac{31}{33}+\varepsilon)n$ admits a $K_4$-decomposition.

Progress towards generalized Nash-Williams' conjecture on $K_4$-decompositions

Abstract

A -decomposition of a graph is a partition of its edges into s. A fractional -decomposition is an assignment of a nonnegative weight to each in a graph such that the sum of the weights of the s containing any given edge is one. Formulating a nonlinear programming and reducing the number of variables slowly, we prove that every graph on vertices with minimum degree at least has a fractional -decomposition. This improves a result of Montgomery that the same conclusion holds for graphs with minimum degree at least . Together with a result of Barber, Kühn, Lo, and Osthus, this result implies that for all , every large enough -divisible graph on vertices with minimum degree at least admits a -decomposition.

Paper Structure

This paper contains 10 sections, 36 theorems, 130 equations.

Table of Contents

  1. Introduction
  2. Fractional $K_4$-decompositions
  3. Formulate a nonlinear programming
  4. The common neighbor density
  5. Main programming
  6. Solving the programming
  7. Reduction to seven variables
  8. Reduction to four variables
  9. Proof of Theorem \ref{["thm:W'_G(O)<=1"]}
  10. Concluding remarks

Key Result

Theorem 1.2

GKLMO19 Let $\varepsilon> 0$ and $F$ be a graph with chromatic number $\chi=\chi(F)\geqslant3$. There exists a constant $N$ such that every $F$-divisible graph $G$ on $n>N$ vertices with minimum degree admits an $F$-decomposition.

Theorems & Definitions (66)

  • Conjecture 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Lemma 2.1
  • proof
  • Theorem 2.2
  • proof
  • ...and 56 more