On a conjecture concerning \(k\)-Hamilton-nice sequences (Q1899021)

The main result of this paper is a proof of a theorem which was first conjectured by Chen and Schelp (personal communication). The concept of \(k\)-Hamilton-nice sequence is due to Liu et al. In the paper under review, the authors prove that a non-negative rational number sequence \((a_1, a_2,\dots, a_{k+ 1})\) is \(k\)-Hamilton-nice if (1) \(a_{k+ 1}\leq 2\), and \[ \sum^h_{j= 1} (i_j- 1)\leq k- 1\quad\text{implies}\quad \sum^h_{j= 1} (a_{i_j}- 1)\leq 1\tag{2} \] for arbitrary \(i_1, i_2,\dots, i_n\in \{1, 2,\dots, k\}\). The result generalizes several known sufficient conditions for graphs to be Hamiltonian.