Edge Colorings Of Graphs With Some Restricted Conditions

Adjacent vertex distinguishing edge coloring,adjacent sum distinguishing edge coloring and twin edge coloring,which are three important concepts of edge coloring with restrictive conditions,they are proper edge colorings that can induce proper vertex coloring by color set,color sum and module color sum,respectively.According to these three induction methods,we define three more general edge colorings with restricted conditions as follows:color set??,??-edge coloring,color sum??,??-edge coloring and module color sum??,??-edge coloring,where?and?are two positive integers.The three concepts of edge coloring with restrictive conditions are called generalized??,??-edge coloring.k-?-distance edge colorings of a graph G that can induce?-distance vertex coloring of G by color set,which is called k-color set??,??-edge coloring of graph G.The minimum k is called color set??,??-edge chromatic number of graph G,defined by ind_a,?^s?G?.k-?-distance edge colorings of a graph G that can induce ?-distance vertex coloring of G by color sum,which is called k-color sum ??,??-edge coloring of graph G.The minimum k is called color sum ??,??-edge chromatic number k of graph G,defined by ind_a,?^t?G?,where the color sets is[k].k-?-distance edge colorings of a graph G that can induce ?-distance vertex coloring of G by module color sum,which is called k-module color sum??,??-edge coloring of graph G.The minimum k is called module color sum??,??-edge chromatic number of graph G,defined by ind_a,?^m?G?,where the color sets is,0{,1...,k-}1.We study the problem of generalized??,??-edge coloring for special graphs and its operation,where ?=?,and ?=1 or ? is a positive even.The details are as follows:1.We obtained that the generalized?2k,2k?-edge chromatic number of path and cycle with order n when ??? and ?n?_2k+1=0.And given generalized ?2,2?-edge chromatic number of path and cycle with order n when k=1.2.We given the generalized ?1,1?-edge chromatic number of the Cartesian product of two paths,and obtained the generalized?2,2?-edge chromatic number of the semistrong product of two paths of order 2 and 3n respectively.3.We obtained the generalized ?1,1?-edge chromatic numbers of the Cartesian product and the direct product of infinite paths.4.We proven that the color set ?1,1?-edge chromatic number and the color sum?1,1?-edge chromatic number of the join with two graph P_n?or C_n? and K_n are equal to 2n.5.We given an upper bound on the color sum ?1,1?-edge chromatic number for the corona of two graphs,and proven that the upper bound is attained precisely for the corona of two isomorphic regular graphs.