The Adjacent-Vertex Distinguishing Edge Coloring And The Fractional Coloring Of Graphs

As a active research field in Graph theory, coloring of graphs has a variety of applications in combinatorics, electrical engineering and so on. Many new coloring problems are proposed and studied, such as the adjacent-vertex distinguishing edge coloring proposed by Zhongfu Zhang et al. In [1], where he obtained the adjacent-vertex distinguishing edge coloring chromatic number of cycles and complet graphs and so on. In this work, we first discuss upper bounds of the adjacent-vertex distinguishing edge coloring chromatic number of the Cartesian product of graphs, and we obtain the adjacent-vertex distinguishing edge coloring chromatic number of C_m×C_n. In what follows, we consider the fractional coloring of graphs. We obtain a property of Kneser graph, which implies that Stahl's conjecture is wrong. Finally, we put forward three new notions of fractional edge coloring , which are based on a:b coloring. what's more, we prove that the fractional adjacent-vertex distinguishing edge coloring chromatic number and the fractional D(β)-adjacent-vertex distinguishing edge coloring chromatic number and the fractional vertex distinguishing edge coloring chromatic number and the fractional edge coloring chromatic number are equal.