January 2010
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56 Reads
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7 Citations
Journal of Computational and Applied Mathematics
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Publications (2)
January 2009
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42 Reads
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5 Citations
Congressus Numerantium
A modular k-coloring, k≥2, of a graph G without isolated vertices is a coloring of the vertices of G with the elements in ℤ k (where adjacent vertices may be colored the same) having the property that for every two adjacent vertices of G, the sums of the colors of their neighbors are different in ℤ k . The minimum k for which G has a modular k-coloring is the modular chromatic number mc(G) of G. The modular chromatic number of a graph is at least as large as its chromatic number. It was known that if T is a nontrivial tree, then mc(T)=2 or mc(T)=3. A nontrivial tree T is of type one if mc(T)=2 and is of type two if mc(T)=3. It is shown that all nontrivial trees of diameter at most 6 are of type one. A caterpillar is a tree of order 3 or more, the removal of whose end-vertices produces a path. A characterization has been established for all caterpillars that are of type two.
Citations (2)
... For a simple undirected and non -trivial connected graph G (V, E), the modular coloring was initiated by F. Okamoto, E. Salehi, and P. Zhang (1) . The modular coloring for some standard graphs is discussed (1,2) . ...
- Citing Article
January 2010
Journal of Computational and Applied Mathematics
... Pewarnaan modular pada graf adalah pemetaan ∶ ( ) → ℤ ( ≥ 2) memungkinkan setiap titik berdekatan memiliki warna yang sama dan ( ) ≠ ( ) untuk titik , yang bertetangga di . Dalam studi Okamoto, Salehi dan Zhang tentang tentang pewarnaan modular pada graf ulat mencapai beberapa hasil yang terkait dengan pewarnaan modular (Okamoto et al., 2009), kemudian Ajiji dan Rahadjeng melakukan penelitian tentang pewarnaan modular pada berbagai subkelas graf yaitu graf sikel, gear, persahabatan dan lintasan (Ajiji & Rahadjeng, 2020). Artikel ini akan mengkaji pewarnaan modular pada beberapa kelas graf pohon: sapu, biner lengkap, bintang ganda, dan pohon pisang. ...
- Citing Article
January 2009
Congressus Numerantium