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On the k-spanning cyclability of 4-valent Cayley graphs on Abelian groups

Brian Alspach, Aditya Joshi

Abstract

A graph $X$ is $k$-spanning cyclable if for any subset $S$ of $k$ distinct vertices there is a 2-factor of $X$ consisting of $k$ cycles such that each vertex in $S$ belongs to a distinct cycle. In this paper we examine the $k$-spanning cyclability of 4-valent Cayley graphs on Abelian groups.

On the k-spanning cyclability of 4-valent Cayley graphs on Abelian groups

Abstract

A graph is -spanning cyclable if for any subset of distinct vertices there is a 2-factor of consisting of cycles such that each vertex in belongs to a distinct cycle. In this paper we examine the -spanning cyclability of 4-valent Cayley graphs on Abelian groups.
Paper Structure (6 sections, 15 theorems, 1 equation, 7 figures)

This paper contains 6 sections, 15 theorems, 1 equation, 7 figures.

Table of Contents

  1. Introduction
  2. The 3-valent case
  3. Graph structure
  4. The product of cycles case
  5. The special case
  6. The circulant case

Key Result

Theorem 2.1

If $X$ is a connected 3-valent Cayley graph on an Abelian group, then $X$ is 2-spanning cyclable if and only if it is isomorphic to $Q_3$ or $K_2\Box C_n$, where $n\geq 4$

Figures (7)

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  • ...and 2 more figures

Theorems & Definitions (31)

  • Definition 1.1
  • Theorem 2.1
  • proof
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • Lemma 4.3
  • proof
  • Theorem 4.4
  • ...and 21 more