A (2, 3)-knight's move on the m × n cylinder chessboard is the move of the knight 2 squares vertically or 2 squares horizontally and then 3 squares perpendicular to it. In this paper, we show a closed (2, 3)-knight's tour on the 5k × n cylinder chessboard for all positive integers k and n, and a closed (2, 3)-knight's tour on the 9k × n cylinder chessboard for all positive integers k and n ∈ {4,

The circumference of a graph G is the length of a longest cycle in G , denoted by cir G . For any even number n , let c n  = min { cir G | G is a 3-connected cubic triangle-free plane graph with n vertices}. In this paper, we show that an upper bound of c n is n + 1 − 3 ⌊ n / 136 ⌋ for n ≥ 136 .