In this note, we prove that a graph is of class one if \(G\) can be embedded in a surface with positive characteristic and satisfies one of the following conditions:(i) \(\Delta(G) \geq 3\) and \(g(G)\)(the girth of \(G\)) \(\geq 8\) (ii) \(\Delta(G) \geq 4\) and \(g(G) \geq 5\)(iii) \(\Delta(G) \geq 5\) and \(g(G) \geq 4\).