We prove that every planar graph with maximum degree ∆ is strong edge (2∆−1)-colorable if its girth is at least 40 [...] +1. The bound 2∆−1 is reached at any graph that has two adjacent vertices of degree ∆.
An edge-colored graph is proper connected if every pair of vertices is connected by a proper path. The proper connection number of a connected graph , denoted by , is the smallest number of colors that are needed to color the edges of in order to make it proper connected. In this paper, we obtain the sharp upper bound for of a general bipartite graph and a series of extremal graphs. Additionally, we give a proper -coloring for a connected bipartite graph having and a dominating cycle...