Under an arc-coloring c of a digraph D , if for each pair of vertices (u, v ) , there exists a directed walk from u to v satisfying that any two consecutive arcs of it have different colors, we say that D is properly-walk connected, and c is a proper-walk coloring of D . The proper-walk connection number (wc) over right arrow of D is the least integer k such that D has a proper-walk coloring with k colors. Fiedorowicz, Sidorowicz, Sopena (Appl. Math. Comput. 410 (2021) 126253) conjectured that if D is a hamiltonian digraph with delta (D) >= 2, then (wc) over right arrow (D) <= 2 . In this paper, we disprove the conjecture by constructing two families of counterexamples. Also, we present some cases for a hamiltonian digraph D having (wc) over right arrow (D ) = 2 . In addition, we find that Observation 15 is not true in the same paper. (C) 2022 Elsevier Inc. All rights reserved.
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