• Journal of applied mathematics & informatics
• /
• v.40 no.1_2
• /
• pp.117-132
• /
• 2022

Reverse edge magic(REM) labeling of the graph G = (V, E) is a bijection of vertices and edges to a set of numbers from the set, defined by λ : V ∪ E → {1, 2, 3, …, |V| + |E|} with the property that for every xy ∈ E, constant k is the weight of equals to a xy, that is λ(xy) - [λ(x) + λ(x)] = k for some integer k. We given the construction of REM labeling for the Cartesian Product, Unions of Braids and Unions of Triangular Belts. The Kotzig array used in this paper is the 3 × (2r + 1) kotzig array. we test the konow results about REM labelling that are related to the new results we found.

• Bulletin of the Korean Mathematical Society
• /
• v.59 no.3
• /
• pp.617-642
• /
• 2022

For positive integers n and d with d < n, an n × n array A based on 𝒳 = {0, 1, …, nd} is called a sparse anti-magic square of order n with density d, denoted by SAMS(n, d), if each non-zero element of X occurs exactly once in A, and its row-sums, column-sums and two main diagonal-sums constitute a set of 2n + 2 consecutive integers. An SAMS(n, d) is called regular if there are exactly d non-zero elements in each row, each column and each main diagonal. In this paper, we investigate the existence of regular sparse anti-magic squares of order n ≡ 1, 5 (mod 6), and prove that there exists a regular SAMS(n, d) for any n ≥ 5, n ≡ 1, 5 (mod 6) and d with 2 ≤ d ≤ n - 1.