Infinitely Many Trees with Maximum Number of Holes Zero, One, and Two
Srinivasa Rao Kola (),
Balakrishna Gudla () and
P. K. Niranjan ()
Journal of Applied Mathematics, 2018, vol. 2018, 1-14
An -coloring of a simple connected graph is an assignment of nonnegative integers to the vertices of such that if and if for all , where denotes the distance between and in . The span of is the maximum color assigned by . The span of a graph , denoted by , is the minimum of span over all -colorings on . An -coloring of with span is called a span coloring of . An -coloring is said to be irreducible if there exists no -coloring g such that for all and for some . If is an -coloring with span , then is a hole if there is no such that . The maximum number of holes over all irreducible span colorings of is denoted by . A tree with maximum degree having span 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.
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