Recent results on the adjacent vertex distinguishing chromatic index of the direct product of graphs
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Direct product
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A proper total-coloring is called vertex-distinguishing total-coloring if the color sets of different vertex and its incident edge are different.The minimum number of colors of a vertex-distinguishing total-coloring is called the vertex-distinguishing total chromatic numbers of the graph.In this paper,the vertex-distinguishing total chromatic numbers of Sm□Sn,Sm□Fnand Fm□Fnare given.
Cartesian product
Total coloring
Fractional coloring
Brooks' theorem
Edge Coloring
Neighbourhood (mathematics)
Complete coloring
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Coloring problem is a classical difficult problem of graph theory. It is a fundamental problem in scientific computation and engineering design. In recent years, a variety of graph coloring problems frequently appeared and solved many problems in production. It is a difficult problem to discuss the chromatic number of a given graph class. In the paper, we introduce several kinds of chromatic numbers of graphs such as adjacent-vertex-distinguishing total chromatic number, adjacent-vertex-distinguishing proper edge chromatic number, smarandachely-adjacent-vertex-distinguishing edge chromatic number, and the multi-fan graphs are considered.
Windmill graph
Graph Coloring
Brooks' theorem
Edge Coloring
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Brooks' theorem
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Brooks' theorem
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Edge Coloring
Windmill graph
Brooks' theorem
Friendship graph
Critical graph
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A proper total coloring of the graph G is called vertex-distinguishing total coloring,if any two vertices have different color sets,where the color set of a vertex is the set composed of all colors of the vertex and the edges incident to it.On the base of the bound of vertex-distinguishing total chromatic number(χvt(G)≤|V(G)|+2).The new upper bound of vertex-distinguishing total chromatic number is obtained by way of probability.
Fractional coloring
Brooks' theorem
Neighbourhood (mathematics)
Complete coloring
Total coloring
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The Adjacent-Vertex Distinguishing Edge Chromatic Number of Some Classes of Cartesian Product Graphs
Let G be a simple connected graph with ordr not less than 3,k-proper edge coloring of G is called adjacent-vertex distinguishing,if for arbitrary two adjacent vertices which are incident to different sets of colored edges.The minimum number required for an adjacent-vertex distinguishing edge coloring(AVDEC) of G is called the adjacent strong edge chromatic number.In this paper,we give two upper bounds of adjacent-vertex distinguishing edge chromatic number of Cartesian product graphs,and some results are obtained.
Cartesian product
Edge Coloring
Neighbourhood (mathematics)
Brooks' theorem
Simple graph
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Windmill graph
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