The connection between the magical coloring of trees
Let $ f $ be a set-ordered edge-magic labeling of a graph $ G $ from $ V(G) $ and $ E(G) $ to $ [0, p-1] $ and $ [1, p-1] $, respectively; it also satisfies the following conditions: $ |f(V(G))| = p $, $ \max f(X) < \min f(Y) $, and $ f(x)+f(y)+f(xy) = C $ for each edge $ xy\in E(G) $. In thi...
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| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
AIMS Press
2024-09-01
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| Series: | AIMS Mathematics |
| Subjects: | |
| Online Access: | https://www.aimspress.com/article/doi/10.3934/math.20241354?viewType=HTML |
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| Summary: | Let $ f $ be a set-ordered edge-magic labeling of a graph $ G $ from $ V(G) $ and $ E(G) $ to $ [0, p-1] $ and $ [1, p-1] $, respectively; it also satisfies the following conditions: $ |f(V(G))| = p $, $ \max f(X) < \min f(Y) $, and $ f(x)+f(y)+f(xy) = C $ for each edge $ xy\in E(G) $. In this paper, we removed the restriction that the labeling of vertices could not be repeated, and presented the concept of magical colorings including edge-magic coloring, edge-difference coloring, felicitous-difference coloring, and graceful-difference coloring. We studied the magical colorings on the tree and proved the existence of four kinds of magical colorings on the tree from a set-ordered edge-magic labeling. Further, we revealed the transformation relationship between these kinds of colorings. |
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| ISSN: | 2473-6988 |