On the Vertex-Distinguishing Proper Edge-Colorings of Graphs

Point-distinguishing chromatic index of the union of paths

“…The remaining unions of three paths can be packed into K 0 6 as follows: 1, 6, 3, 5, 6, 2, 2), (3, 3, 1, 2, 5, 4, 4), (5, 5, 1, 4, 6, 6), P 9 ∪ P 6 ∪ P 6 → (1, 1, 2, 6, 5, 3, 4, 2, 2), (3, 3, 2, 5, 4, 4), (5, 5, 1, 4, 6, 6), P 7 ∪ P 6 ∪ P 6 → (1, 1, 6, 3, 4, 2, 2), (3, 3, 2, 5, 4, 4), (5, 5, 1, 4, 6, 6), P 9 ∪ P 5 ∪ P 5 → (1, 1, 5, 4, 6, 5, 3, 2, 2), (3, 3, 1, 4, 4), (5, 5, 2, 6, 6), P 8 ∪ P 6 ∪ P 5 → (1, 1, 6, 5, 3, 4, 2, 2), (3, 3, 1, 5, 4, 4), (5, 5, 2, 6, 6), P 7 ∪ P 7 ∪ P 5 → (1, 1, 6, 3, 4, 2, 2), (3, 3, 1, 5, 6, 4, 4), (5, 5, 2, 6, 6), P 8 ∪ P 7 ∪ P 4 → (1, 1, 5, 2, 4, 6, 2, 2), (3, 3, 1, 6, 3, 4, 4), (5,5,6,6), P 9 ∪ P 6 ∪ P 4 → (1, 1, 6, 2, 5, 4, 3, 2, 2), (3, 3, 1, 2, 4, 4), (5,5,6,6), P 10 ∪ P 5 ∪ P 4 → (1, 1, 6, 2, 5, 1, 4, 3, 2, 2), (3,3,5,4,4), (5,5,6,6), P 11 ∪ P 4 ∪ P 4 → (1, 1, 6, 2, 3, 1, 2, 5, 4, 2, 2), (3,3,4,4), (5,5,6,6), P 6 ∪ P 6 ∪ P 6 → (1, 1, 3, 6, 2, 2), (3, 3, 2, 5, 4, 4), (5, 5, 1, 4, 6, 6), P 8 ∪ P 5 ∪ P 5 → (1, 1, 5, 6, 4, 3, 2, 2), (3, 3, 1, 4, 4), (5, 5, 2, 6, 6), P 7 ∪ P 6 ∪ P 5 → (1, 1, 6, 5, 3, 2, 2), (3, 3, 1, 5, 4, 4), (5, 5, 2, 6, 6), P 7 ∪ P 7 ∪ P 4 → (1, 1, 6, 4, 5, 2, 2), (3, 3, 4, 1, 2, 4, 4), (5,5,6,6), (1, 1, 6, 3, 4, 5, 2, 2), (3, 3, 2, 1, 4, 4), (5,5,6,…”

The Vertex-Distinguishing Edge Coloring of 𝑷𝒎⋁𝑲𝒏 and 𝑪𝒎 ∨ 𝑲𝒏

“…′ ( ) = { | − } Definition 2 [1][2][3][4] For the proper edge coloring of simple graph, if it is satisfied with ( ) ≠ ( ) for ( )( ≠ ) , where ( ) = { ( )| ∈ ( )} , then is called the Vertex-distinguishing Edge Coloring, is abbreviated − of , and is called the…”

On the Vertex-Distinguishing Edge Chromatic Number of 𝐏𝐦 ∨ 𝐂𝐧

“…The problem about vertex-distinguishing edge coloring of is a widely used and extremely difficult problem [1][2][3][4] . In [5] introduced the vertex-distinguishing edge coloring of graph, and give the correspondence conjecture.…”