Abstract. Let ( , ) be a simple graph, is a mapping from ( ) βͺ ( ) to {1,2, β― , }. Let ( ) = { ( )} βͺ { ( )| β ( ), β ( )} for every β ( ) . If is a K-proper-total-coloring, and for β , β ( ) , we have ( ) β ( ) , then is called the k-vertex-distinguishing total coloring (k-VDTC for shot). Let β² ( ) = min { | has a k-vertex-distinguishing total colorint}. Then β² ( ) is called the vertex-distinguishing total chromatic number. The total chromatic number on β¨ and β .
Abstract.A proper edge coloring of graph G is called equitable adjacent strong edge coloring if colored sets from every two adjacent vertices incident edge are different, and the number of edges in any two color classes differ by at most one, which the required minimum number of colors are called the adjacent strong equitable edge chromatic number. In this paper, we obtain vertex-distinguishing edge coloring of β¨ and β¨ .
Abstract.A proper edge coloring of graph G is called equitable adjacent strong edge coloring if colored sets from every two adjacent vertices incident edge are different, and the number of edges in any two color classes differ by at most one, which the required minimum number of colors is called the adjacent strong equitable edge chromatic number. In this paper, we present the edge coloring of join-graphs about path and cycle, and gain the vertex-distinguishing edge chromatic number of β¨ .
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