Classification of edge‐transitive rose window graphs

Given natural numbers n?3 and 1?a, r?n?1, the rose window graph R_n(a, r) is a quartic graph with vertex set ${{{x}}_{{i}}|{{i}}in {mathbb{Z}}_{{n}}} cup {{{y}}_{{i}}|{{i}}in{mathbb{Z}}_{{n}}}Given natural numbers n?3 and 1?a, r?n?1, the rose window graph R_n(a, r) is a quartic graph with vertex set ${{{x}}_{{i}}|{{i}}in {mathbb{Z}}_{{n}}} cup {{{y}}_{{i}}|{{i}}in{mathbb{Z}}_{{n}}}$ and edge set ${{{{x}}_{{i}},{{x}}_{{{i+1}}}} mid {{i}}in {mathbb{Z}}_n } cup {{{{y}}_{{{i}}},{{y}}_{{{i+r}}}}mid {{i}} in{mathbb{Z}}_{{n}}}cup {{{{x}}_{{{i}}},{{y}}_{{{i}}}} mid {{i}}in {mathbb{Z}}_{{{n}}}}cup {{{{x}}_{{{i+a}}},{{y}}_{{{i}}}} mid{{i}} in {mathbb{Z}}_{{{n}}}}$. In this article a complete classification of edge‐transitive rose window graphs is given, thus solving one of the three open problems about these graphs posed by Steve Wilson in 2001. © 2010 Wiley Periodicals, Inc. J Graph Theory 65: 216–231, 2010