Journal of Applied Mathematics

Volume 2018, Article ID 8186345, 14 pages

https://doi.org/10.1155/2018/8186345

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

Department of Mathematical and Computational Sciences, National Institute of Technology Karnataka, Surathkal, India

Correspondence should be addressed to Srinivasa Rao Kola; moc.liamg@pgktii.unirs

Received 28 April 2018; Accepted 18 August 2018; Published 20 September 2018

Academic Editor: Ali R. Ashrafi

Copyright © 2018 Srinivasa Rao Kola et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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.

#### 1. Introduction

The channel assignment problem is the problem of assigning frequencies to transmitters in some optimal manner. In 1992, Griggs and Yeh [1] have introduced the concept of -coloring as a variation of channel assignment problem. The distance between two vertices and in a graph , denoted by , is defined as the length of a shortest path between and in . An -coloring of a graph is an assignment such that, for every in , if and are adjacent and if and are at distance 2. The nonnegative integers assigned to the vertices are also called colors. The span of , denoted by , is . The span of , denoted by , is . An -coloring with span is called a span coloring. A tree is a connected acyclic graph. In the introductory paper, Griggs and Yeh [1] proved that for ; is either or for any tree with maximum degree . We refer to a tree as Type-I if ; otherwise it is Type-II. In a graph with maximum degree , we refer to a vertex as a major vertex if its degree is ; otherwise is a minor vertex. Wang [2] has proved that a tree with no pair of major vertices at distances 1, 2, and 4 is Type-I. Zhai et al. [3] have improved the above condition as a tree with no pair of major vertices at distances 2 and 4 is Type-I. Mandal and Panigrahi [4] have proved that if has at most one pair of major vertices at distance either 2 or 4 and all other pairs are at distance at least 7. Wood and Jacob [5] have given a complete characterization of the -span of trees up to twenty vertices.

Fishburn and Roberts [6] have introduced the concept of no-hole -coloring of a graph. If is an -coloring of a graph with span , then an integer is called a hole in if there is no vertex in such that . An -coloring with no hole is called a no-hole coloring of . Fishburn et al. [7] have introduced the concept of irreducibility of -coloring. An -coloring of a graph is reducible if there exists another -coloring of such that for all vertices in and there exists a vertex in such that . If is not reducible then it is called irreducible. An irreducible no-hole coloring is referred to as inh-coloring. A graph is inh-colorable if there exists an inh-coloring. For an inh-colorable graph , the lower inh-span or simply inh-span of , denoted by , is defined as . Fishburn et al. [7] have proved that paths, cycles, and trees are inh-colorable except , , and stars. In addition to that, they showed that where is any nonstar tree. Laskar et al. [8] have proved that any nonstar tree is inh-colorable and . The maximum number of holes over all irreducible span colorings of is denoted by . Laskar and Eyabi [9] have determined the exact values for maximum number of holes for paths, cycles, stars, and complete bipartite graphs as 2, 2, 1, and 1, respectively, and conjectured that, for any tree , if and only if is a path . S. R. Kola et al. [10] have disproved the conjecture by giving a two-hole irreducible span coloring for a Type-II tree other than path.

In this article, we give a method of construction of infinitely many two-hole trees from a two-hole tree and infinitely many trees with at least one hole from a one-hole tree. Also, we find maximum number of holes for some Type-II trees given by Wood and Jacob [5] and obtain infinitely many Type-II trees of holes one and two by applying the method of construction. Further, we give a sufficient condition for a zero-hole Type-II tree.

#### 2. Construction of Trees with Maximum Number of Holes One and Two

We start this section with a lemma which gives the possible colors to the major vertices in a two-hole span coloring of a Type-II tree.

Lemma 1. *In any two-hole span coloring of a Type-II tree with , all major vertices receive either the same color or the colors from any one of the sets , , or .*

*Proof. *Let be a two-hole span coloring of a Type-II tree . Suppose that and are major vertices such that . First, we prove that or or . Let and . Without loss of generality, we assume that . If , then the color 1 must be one of the two holes in . If , then and are the holes. Since cannot be 1, is 1 which implies . If , then and are the holes in . If , then and are the holes which are not possible as . If , then must be one of the holes in . Since cannot be , is which implies .

If , then 1 and 3 are the holes. If any major vertex receives a color other than 0 and 2, then the neighbors of cannot get the colors 1 and 3 and at least one of and . This is not possible as we need number of colors to color a major vertex and its neighbors. Similarly, other cases can be proved.

The following lemma is a direct implication of Lemma 1.

Lemma 2. *If is a two-hole span coloring of a Type-II tree having two major vertices at distance less than or equal to two, then the set of holes in is , , or .*

When we say connecting two trees, we mean adding an edge between them. Corresponding to the possibilities of holes given in Lemma 2, we give a list of trees which can be connected to a two-hole tree having two major vertices at distance less than or equal to two, to obtain infinitely many two-hole trees. Later, we give a list of trees which can be connected to a one-hole tree to get infinitely many one-hole trees.

Theorem 3. *If is a tree with maximum number of holes two and having at least two major vertices at distance at most two, then there are infinitely many trees with maximum number of holes two and with maximum degree same as that of .*

*Proof. *Let be an irreducible span coloring of with two holes. Then by Lemma 2, the set of holes in is or or . Now, we give a method to construct trees from using the coloring and holes in . For all the three possibilities of holes, we give a list of trees which can be connected to to get a bigger tree with maximum number of holes two. Suppose and are the holes in . We use Table 1 for construction.

Let be a vertex of the tree and be the color received by . Now depending on the colors of the neighbors of , to preserve -coloring, we connect the trees (one at a time) given in Table 1 by adding an edge between of and the vertex colored of tree in the table. Note that and the color is not equal to any of the colors , , , , and and not assigned to any neighbor of . To maintain irreducibility, we use the condition given in the last column of the table. It is easy to see that, after every step, we get a tree with maximum degree same as that of and a two-hole irreducible span coloring of . Also, it is clear that is a subtree of . Since connecting a tree to any pendant vertex is always possible, we get infinitely many trees.

Suppose and are the holes in . Construction is similar to the previous case using trees in Table 2.

Suppose and are the holes in . We use trees in Table 3 for construction.