The geodesic curvature flow is an important concept in Riemannian geometry. The flow with level set formulation has many applications in image processing, computer vision, material sciences, etc. The existing discretizations on triangulated surfaces are based on either finite volume method or finite element method with piecewise linear function space, which are suitable for vertex-based two-phase problems. The contour (zero level set) in existing methods passes through triangles of the mesh. However, in some graphic applications, such as mesh segmentation (to divide a whole mesh into several sub-meshes without ambiguous triangular stripes), the cutting contour is needed to be along the edges of the mesh. Moreover, multi-phase segmentation by a single level set function is a difficult problem for a long time. In this paper, we try to tackle these two problems. We propose a new discretization which has simpler formulation and more sparse coefficient matrix. We prove the existence and uniqueness, regularization behavior and maximum-minimum principle of our discrete flow. Therein the maximum-minimum principal has not been presented before. Lots of experiments show that, the limit of the flow would be a piecewise constant solution with 'discontinuity set' to be the closed geodesics of the surface. We therefore propose a constrained discrete geodesic curvature flow, which is also analyzed theoretically. The linear system of the constrained flow can be equivalently reformulated into a much smaller one (especially in the narrow band algorithm), which dramatically reduces the computation cost. Combined with a narrow band algorithm, the constrained flow with topologically correct initializations (easy to be got by simple existing methods or manual inputs) yields a multi-phase segmentation method by a single level set function. We test our two flows in closed curve evolution and multi-region segmentation applications. The numerical experiments are given to demonstrate the effectiveness.