  In a series of papers, of which this is the first, we study sufficient
conditions for Hamiltonicity in terms of forbidden induced subgraphs and extend
such results to locally finite infinite graphs. For this we use topological
circles within the Freudenthal compactification of a locally finite graph as
infinite cycles. In this paper we focus on conditions involving claws, nets and
bulls as induced subgraphs. We extend Hamiltonicity results for finite
claw-free and net-free graphs by Shepherd to locally finite graphs. Moreover,
we generalise a classification of finite claw-free and net-free graphs by
Shepherd to locally finite ones. Finally, we extend to locally finite graphs a
Hamiltonicity result by Ryj\'{a}\v{c}ek involving a relaxed condition of being
bull-free.
