A construction of smooth travel groupoids on finite graphs

A travel groupoid is an algebraic system related with graphs. In this paper, we give an algorithm to construct smooth travel groupoids for any finite graph. This algorithm gives an answer of L.~Nebesk$\acute{\mbox{y}}$'s question, "Does there exists a connected graph $G$ such that $G$ has no smooth travel groupoid?", in finite cases.

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