All generalized rose window graphs are hamiltonian

Abstract A bicirculant is a regular, d -valent graph that admits a semiregular automorphism of order m having two vertex-orbits of size m . The vertices of each orbit induce a circulant graph of order m and the remaining edges span a regular bipartite graph of valence, say s , $$1 \le s \le d$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>≤</mml:mo> <mml:mi>s</mml:mi> <mml:mo>≤</mml:mo> <mml:mi>d</mml:mi> </mml:mrow> </mml:math> , connecting the two vertex-orbits. Generalized Petersen graphs constitute a prominent family of bicirculants, with $$d = 3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>=</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> and $$s = 1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>s</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> . In 1983, Brian Alspach proved that all generalized Petersen graphs are hamiltonian, except for the family G ( m , 2) with $$m\equiv 5\pmod 6$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>≡</mml:mo> <mml:mn>5</mml:mn> <mml:mspace/> <mml:mo>(</mml:mo> <mml:mo>mod</mml:mo> <mml:mspace/> <mml:mn>6</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . In this paper we conjecture that among all connected bicirculants of valence at least 2, there are no other exceptions. It follows from various sources that the conjecture is true for all cubic bicirculants. In this paper we prove the conjecture for quartic bicirulants with $$s = 2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>s</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> , also known as the generalized rose window graphs.