Cayley Graphs Defined by Systems of Equations

Fuyuan Yang,Chao Zhang,Qiang Sun,Hongbo Zhou

doi:10.3390/axioms11030100

Abstract

Let R be a finite ring. In this paper, we mainly explore the conditions to ensure the graph BΓn defined by a system of equations {fi|i=2,…,n} to be a Cayley graph or a Hamiltonian graph. More precisely, we prove that BΓn is a Cayley graph with G=⟨ϕ,A⟩ a group of dihedral type if and only if the system Fn={fi|i=2,…,n} is Cayley graphic of dihedral type in R. As an application, the well-known Lova´sz Conjecture, which states that any finite connected Cayley graph has a Hamilton cycle, holds for the connected BΓn defined by Cayley graphic system Fn of dihedral type in the field GF(pk).

Highlights

The graph BΓn(R; f2, . . . , fn) defined by a system of equations over a finite ring R have been extensively studied and used since the 1990s
The construction of the graphs by systems of equations provides a useful tool to study graph theory, for instance, constructing the graphs D(k, q) and CD(k, q), to approximate the structure and behavior of incidence graphs of regular generalized polygons [4], establishing the lower bound of the Ramsey numbers of graphs associated to generalized Kac–Moody algebras of rank two [5] and hypergraphs [6], and constructing certain graphs with given properties [1,7,8]
The present paper aims to study the question of when BΓn(R; f2, . . . , fn) is a Cayley graph

Summary

Introduction

The graph BΓn(R; f2, . . . , fn) defined by a system of equations over a finite ring R have been extensively studied and used since the 1990s. The graph BΓn is a Cayley graph on some group G = φ, A of dihedral type with φ : (p) ↔ [p] ∈ Aut(BΓn) and A uniform if and only if the function system Fn = { fi | i = 2, . For Cayley graphic function system of dihedral type, we always take uniform automorphisms X = {σk,x | x ∈ R, k = 1, .

Results

Conclusion