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Nowhere-zero $3$-flows in Cayley graphs on solvable groups of twice square-free order

Milad Ahanjideh, István Kovács

Abstract

We verify Tutte's $3$-flow conjecture in the class of Cayley graphs on solvable groups of order $2n$, where $n$ is square-free. The proof relies on a new necessary and sufficient condition for a simple $5$-valent graph to admit a nowhere-zero $3$-flow in terms of a pseudoforest decomposition.

Nowhere-zero $3$-flows in Cayley graphs on solvable groups of twice square-free order

Abstract

We verify Tutte's -flow conjecture in the class of Cayley graphs on solvable groups of order , where is square-free. The proof relies on a new necessary and sufficient condition for a simple -valent graph to admit a nowhere-zero -flow in terms of a pseudoforest decomposition.
Paper Structure (6 sections, 24 theorems, 53 equations, 1 figure, 1 table)

This paper contains 6 sections, 24 theorems, 53 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Nowhere-zero 3-flows and a pseudoforest decomposition
  4. Proof of Theorem \ref{['main']}
  5. Graphs G1 and G2
  6. Graphs G3 and G4

Key Result

Theorem 1.2

Let $G$ be a solvable group of order $2n$, where $n$ is square-free. Then every connected Cayley graph of valency at least $4$ on $G$ admits a nowhere-zero $3$-flow.

Figures (1)

  • Figure 1: The graph $\Lambda[L]$ (left) and the graph $\Gamma'$ (right) obtained from $\Lambda[L]$.

Theorems & Definitions (45)

  • Conjecture 1.1
  • Theorem 1.2
  • Remark 1.3
  • Proposition 2.1: ZZ
  • Proposition 2.2: NS
  • Proposition 2.3: NS
  • Proposition 2.4: NS
  • Proposition 2.5: NS
  • Proposition 2.6: ZsZ
  • Proposition 2.7: Hbook
  • ...and 35 more