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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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

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

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

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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.

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