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Errata to: Chapter 50 in: J.K. Mandal et al. (eds.), Information Systems Design and Intelligent Applications, Advances in Intelligent Systems and Computing 339, DOI 10.1007/978-81-322-2250-7_50
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Page: 512. The following sentence is required to be added at the end of paragraph 1 in Sect. 1.
“Some results of simple graphs with L(4, 3, 2, 1) labeling can be found in [9]”.
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Page: 513. “Theorem 1” should be read as “Theorem 1 [6]”.
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Page: 514. “Theorem 2” should be read as “Theorem 2 [6]”.
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Page: 515. The Lemma 1 along with its proof in Sect. 3.3 should be read as:
Lemma 1
For a path \( P_{n} \) on \( n \) vertices with \( n\,\ge\,7 \) , the minimal L(4, 3, 2, 1)-labeling number \( \lambda(P_{n} ) \) is at most 13.
Proof
A labeling pattern \( \{ f(v_{1} ),\,f(v_{2} ), \ldots ,\,f(v_{7} )\} = \{ 5,9,13,3,7,11,1\} \) exists for n = 7. Hence the lemma follows.
Page: 515. The Theorem 3 and its proof for Case-IV and Case-V in Sect. 3.3 should be read as:
Theorem 3
For a path, \( P_{n} \) on \( n \) vertices, the minimal L(4, 3, 2, 1)-labeling number \( \lambda(P_{n} ) \) is
Proof
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Case-IV: \( n = 4\):
The labeling pattern \( \{ 6,1,9,4\} \) shows that \( \lambda (P_{n} ) \le 9 \) if \( n = 4 \). Let \( V(P_{n} ) = \{ v_{1} ,v_{2} ,v_{3} ,v_{4} \} \). \( V(P_{n} ) \) has two vertices of degree 2 and other two vertices of degree 1. If either \( f(v_{2} ) \) or \( f(v_{3} ) \) is 1 then either \( f(v_{4} ) \) or \( f(v_{1} ) \) will be at least 12, which is a contradiction. Similar contradiction will arrive if either \( f(v_{1} ) \) or \( f(v_{4} ) \) is set to 1.
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Case-V: \( n = 5,6,7\):
Since \( \exists \) a labeling \( \{ 8,3,11,6,1,9,4\} \), we can assume that \( \lambda (P_{n} ) \le 11 \) for \( n = 5,6,7 \). Let \( f(v_{i} ) = 1 \) and either \( v_{i + 1} \), \( v_{i + 2} \) or \( v_{i - 1} \), \( v_{i - 2} \) exist. Now \( \lambda (P_{3} ) = 8 \) implies that \( f(v_{i + 1} ) \) is either 5, 6, 7 or 8. For L(3, 2, 1)-labeling [6], note that the possibilities for \( f(v_{i + 1} ) \) is either 5, 6, 7 or 8. Therfore, the similar approach in [6] can be used to handle this case.
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Page: 517. The Claim 1 is not correct and hence the last line of the “Abstract” should be read as “This paper also presents an L(4, 3, 2, 1)-labeling algorithm for path.”
Reference
Sweetly, R.: A study on radio labeling and related concepts in graphs. PhD thesis, Manonmaniam Sundaranar University (2011)
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Atta, S., Mahapatra, P.R.S. (2015). Errata to: L(4, 3, 2, 1)-Labeling for Simple Graphs. In: Mandal, J., Satapathy, S., Kumar Sanyal, M., Sarkar, P., Mukhopadhyay, A. (eds) Information Systems Design and Intelligent Applications. Advances in Intelligent Systems and Computing, vol 339. Springer, New Delhi. https://doi.org/10.1007/978-81-322-2250-7_88
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DOI: https://doi.org/10.1007/978-81-322-2250-7_88
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