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

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 $\left|f\left(u\right)-f\left(v\right)\right|⩾\mathrm{2}$ if $d(u,v)=\mathrm{1}$ and $\left|f\left(u\right)-f\left(v\right)\right|⩾\mathrm{1}$ 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 $g(u)⩽f(u)$ 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.