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The Lovász conjecture holds for moderately dense Cayley graphs

Benjamin Bedert, Nemanja Draganić, Alp Müyesser, Matías Pavez-Signé

Abstract

We show that there is an absolute constant $c>0$ such that every large connected $n$-vertex Cayley graph with degree $d\geq n^{1-c}$ has a Hamilton cycle. This makes progress towards the Lovász conjecture and improves upon the previous best result of this form due to Christofides, Hladký, and Máthé from 2014 concerning graphs with $d\geq \varepsilon n$. Our proof avoids the use of Szemerédi's regularity lemma and relies instead on an efficient arithmetic regularity lemma specialised to Cayley graphs.

The Lovász conjecture holds for moderately dense Cayley graphs

Abstract

We show that there is an absolute constant such that every large connected -vertex Cayley graph with degree has a Hamilton cycle. This makes progress towards the Lovász conjecture and improves upon the previous best result of this form due to Christofides, Hladký, and Máthé from 2014 concerning graphs with . Our proof avoids the use of Szemerédi's regularity lemma and relies instead on an efficient arithmetic regularity lemma specialised to Cayley graphs.
Paper Structure (14 sections, 18 theorems, 30 equations, 2 figures)

This paper contains 14 sections, 18 theorems, 30 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Future directions
  3. Outline of the proof
  4. First reduction
  5. Well-connected case: Theorem \ref{['thm:maintechnical']}
  6. Preliminaries
  7. Probabilistic tools
  8. Cayley graphs
  9. Linear forests in almost-regular graphs
  10. Proof of Theorem \ref{['thm:mainthm']}
  11. Robust expansion in graphs and digraphs
  12. Proof of Theorem \ref{['thm:maintechnical']}
  13. Non-bipartite case
  14. Bipartite case

Key Result

Theorem 1.2

There exists a constant $c\geq1/200$ such that for all sufficiently large $n$, every connected $n$-vertex Cayley graph of degree $d > n^{1-c}$ has a Hamilton cycle.

Figures (2)

  • Figure 1: A picture of the $v$-absorber $F_5$ and the corresponding $(x_0,y_0)$-paths.
  • Figure 2: This picture shows the path $1234$ in the auxiliary digraph, which then corresponds to a path of absorbers in the original graph. These absorbers combined can incorporate any subset of $\{g_iv:i\in [4]\}$ to the final path. In this picture, a path is shown incorporating $g_1v$ and $g_4v$ while avoiding $g_2v$ and $g_3v$.

Theorems & Definitions (37)

  • Conjecture 1.1
  • Theorem 1.2
  • Conjecture 1.3
  • Definition 2.1
  • Theorem 2.2: Bedert, Bucić, Kravitz, Montgomery, Müyesser bedert2025graham
  • Definition 2.3: Auxiliary graph
  • Theorem 2.4
  • Definition 2.5
  • Lemma 3.1: Chernoff's Bound JLR2000
  • Lemma 3.2: McDiarmid's Inequality McDiarmid1989BoundedDifferences
  • ...and 27 more