## Journal of Applied Mathematics

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

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

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

#### Abstract

An $L(\mathrm{2,1})$-coloring of a simple connected graph $G$ is an assignment $f$ of nonnegative integers to the vertices of $G$ such that $\left|f\left(u\right)-f\left(v\right)\right|\u2a7e\mathrm{2}$ if $d(u,v)=\mathrm{1}$ and $\left|f\left(u\right)-f\left(v\right)\right|\u2a7e\mathrm{1}$ if $d(u,v)=\mathrm{2}$ for all $u,v\in V(G)$, 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 $\lambda (G)$, is the minimum of span over all $L(\mathrm{2,1})$-colorings on $G$. An $L(\mathrm{2,1})$-coloring of $G$ with span $\lambda (G)$ is called a span coloring of $G$. An $L(\mathrm{2,1})$-coloring $f$ is said to be irreducible if there exists no $L(\mathrm{2,1})$-coloring g such that $g(u)\u2a7df(u)$ for all $u\in V(G)$ and $$ for some $v\in V(G)$. If $f$ is an $L(\mathrm{2,1})$-coloring with span $k$, then $h\in \left\{\mathrm{0,1},\mathrm{2},\dots ,k\right\}$ is a hole if there is no $v\in V(G)$ such that $f(v)=h$. The maximum number of holes over all irreducible span colorings of $G$ is denoted by ${H}_{\lambda}(G)$. A tree $T$ with maximum degree $\mathrm{\Delta}$ having span $\mathrm{\Delta}+\mathrm{1}$ 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