Abstract
We prove that a 2-connected graph G of order p is traceable if | N( u)∪ N( v)| − | N( w)∪ N( x)| ⩾ p −1 for all 4-tuples { u, v, w, x} with d( u, v) = d( w, x) = 2 ( u, v, w, x are distinct vertices of G). In addition, we give a short proof of Lindquester's conjecture.
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