Abstract
A simple graph G is k-ordered (respectively, k-ordered hamiltonian), if for any sequence of k distinct vertices v 1 , … , v k of G there exists a cycle (respectively, hamiltonian cycle) in G containing these k vertices in the specified order. In 1997 Ng and Schultz introduced these concepts of cycle orderability and posed the question of the existence of 3-regular 4-ordered (hamiltonian) graphs other than K 4 and K 3 , 3 . Ng and Schultz observed that a 3-regular 4-ordered graph on more than 4 vertices is triangle free. We prove that a 3-regular 4-ordered graph G on more than 6 vertices is square free,and we show that the smallest graph that is triangle and square free, namely the Petersen graph, is 4-ordered. Furthermore, we prove that the smallest graph after K 4 and K 3 , 3 that is 3-regular 4-ordered hamiltonianis the Heawood graph. Finally, we construct an infinite family of 3-regular 4-ordered graphs.
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