Journal of Applied Mathematics

Infinitely Many Trees with Maximum Number of Holes Zero, One, and Two

Srinivasa Rao Kola, Balakrishna Gudla, and P. K. Niranjan

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Abstract

An L ( 2,1 ) -coloring of a simple connected graph G is an assignment f of nonnegative integers to the vertices of G such that f u - f v 2 if d ( u , v ) = 1 and f u - f v 1 if d ( u , v ) = 2 for all u , v V ( G ) , where d ( u , v ) denotes the distance between u and v in G . The span of f is the maximum color assigned by f . The span of a graph G , denoted by λ ( G ) , is the minimum of span over all L ( 2,1 ) -colorings on G . An L ( 2,1 ) -coloring of G with span λ ( G ) is called a span coloring of G . An L ( 2,1 ) -coloring f is said to be irreducible if there exists no L ( 2,1 ) -coloring g such that g ( u ) f ( u ) for all u V ( G ) and g ( v ) < f ( v ) for some v V ( G ) . If f is an L ( 2,1 ) -coloring with span k , then h 0,1 , 2 , , k is a hole if there is no v V ( G ) such that f ( v ) = h . The maximum number of holes over all irreducible span colorings of G is denoted by H λ ( G ) . A tree T with maximum degree Δ having span Δ + 1 is referred to as Type-I tree; otherwise it is Type-II. In this paper, we give a method to construct infinitely many trees with at least one hole from a one-hole tree and infinitely many two-hole trees from a two-hole tree. Also, using the method, we construct infinitely many Type-II trees with maximum number of holes one and two. Further, we give a sufficient condition for a Type-II tree with maximum number of holes zero.

Article information

Source
J. Appl. Math., Volume 2018 (2018), Article ID 8186345, 14 pages.

Dates
Received: 28 April 2018
Accepted: 18 August 2018
First available in Project Euclid: 10 October 2018

Permanent link to this document
https://projecteuclid.org/euclid.jam/1539136834

Digital Object Identifier
doi:10.1155/2018/8186345

Mathematical Reviews number (MathSciNet)
MR3859779

Citation

Kola, Srinivasa Rao; Gudla, Balakrishna; Niranjan, P. K. Infinitely Many Trees with Maximum Number of Holes Zero, One, and Two. J. Appl. Math. 2018 (2018), Article ID 8186345, 14 pages. doi:10.1155/2018/8186345. https://projecteuclid.org/euclid.jam/1539136834


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References

  • J. R. Griggs and R. K. Yeh, “Labeling graphs with a condition at distance 2,” SIAM Journal on Discrete Mathematics, vol. 5, no. 4, pp. 586–595, 1992.
  • W.-F. Wang, “The L(2,1)-labelling of trees,” Discrete Applied Mathematics, vol. 154, no. 3, pp. 598–603, 2006.
  • M.-q. Zhai, C.-h. Lu, and J.-l. Shu, “A note on L(2,1)-labelling of trees,” Acta Mathematicae Applicatae Sinica, vol. 28, no. 2, pp. 395–400, 2012.
  • N. Mandal and P. Panigrahi, “Solutions of some L(2,1)-coloring related open problems,” Discussiones Mathematicae Graph Theory, vol. 36, no. 2, pp. 279–297, 2016.
  • C. A. Wood and J. Jacob, “A complete L(2,1)-span characterization for small trees,” AKCE International Journal of Graphs and Combinatorics, vol. 12, no. 1, pp. 26–31, 2015.
  • P. C. Fishburn and F. S. Roberts, “No-hole L(2,1)-colorings,” Discrete Applied Mathematics, vol. 130, no. 3, pp. 513–519, 2003.
  • P. C. Fishburn, R. C. Laskar, F. S. Roberts, and J. Villalpando, “Parameters of L(2,1)-coloring,” Manuscript.
  • R. C. Laskar, G. L. Matthews, B. Novick, and J. Villalpando, “On irreducible no-hole L(2,1)-coloring of trees,” Networks. An International Journal, vol. 53, no. 2, pp. 206–211, 2009.
  • R. Laskar and G. Eyabi, “Holes in L(2,1)-coloring on certain classes of graphs,” AKCE International Journal of Graphs and Combinatorics, vol. 6, no. 2, pp. 329–339, 2009.
  • S. R. Kola, B. Gudla, and N. P.K., “Some classes of trees with maximum number of holes two,” AKCE International Journal of Graphs and Combinatorics, 2018. \endinput