Let k ≥ 2 be an integer and G be a connected graph of order at least 3. A twin k-edge coloring of G is a proper edge coloring of G that uses colors from ℤ k and that induces a proper vertex coloring on G where the color of a vertex v is the sum (in ℤ k ) of the colors of the edges incident with v. The smallest integer k for which G has a twin k-edge coloring is the twin chromatic index of G and is denoted by χ t ′ ( G ) . In this paper, we determine the twin chromatic indices of circulant graphs C n ( 1 , n 2 ) , and some generalized Petersen graphs such as GP(3s, k), GP(m, 2), and GP(4s, l) where n ≥ 6 and n ≡ 0 (mod 4), s ≥ 1, k ≢ 0 (mod 3), m ≥ 3 and m ∉ {4, 5}, and l is odd. Moreover, we provide some sufficient conditions for a connected graph with maximum degree 3 to have twin chromatic index greater than 3.
Let k ≥ 2 be an integer and G be a connected graph of order at least 3. In this paper, we introduce a new neighbor-distinguishing coloring called twin edge mean coloring. A proper edge coloring of G that uses colors from ℕ k = {0,1,…, k − 1} is called a twin k-edge mean coloring of G if it induces a proper vertex coloring of G such that the color of each vertex υ of G is the average of the colors of the edges incident with υ, and is an integer. The minimum k for which G has a twin k-edge mean coloring is called the twin chromatic mean index of G and is denoted by χ t m ′ ( G ) . First, we establish lower and upper bounds for χ t m ′ ( G ) under general or more specific assumptions. Then we determine the twin chromatic mean indices of paths, cycles, and stars.
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