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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

  • 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]”.

  • Page: 513. “Theorem 1” should be read as “Theorem 1 [6]”.

  • Page: 514. “Theorem 2” should be read as “Theorem 2 [6]”.

  • 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

$$ \lambda (P_{n} ) = \left\{ {\begin{array}{*{20}l} 1 \hfill & {if\;{\text{n = 1}}} \hfill \\ 5 \hfill & {if\;{\text{n = 2}}} \hfill \\ 8 \hfill & {if\;{\text{n = 3}}} \hfill \\ {9} \hfill & {if\;{\text{n = 4}}} \hfill \\ {11} \hfill & {if\;{\text{n}} = 5 ,\, 6 ,\,7} \hfill \\ \end{array} } \right. $$

Proof

  • 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.

  • 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.

  • 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.”