An L (2, 1)-coloring of a simple connected graph G is an assignment of non-negative integers to the vertices of G such that adjacent vertices color difference is at least two, and vertices that are at distance two from each other get different colors. The maximum color assigned in an L (2, 1)-coloring is called span of that coloring. The span of a graph G denoted by λ (G) is the smallest span taken over all L (2, 1)-colorings of G. A hole is an unused color within the range of colors used by the coloring. An L (2, 1)-coloring f is said to be irreducible if no other L (2, 1)-coloring can be produced by decreasing a color of f . The maximum number of holes of a graph G, denoted by H λ (G), is the maximum number of holes taken over all irreducible L (2, 1)-colorings with span λ (G). Laskar and Eyabi (Christpher, 2009) conjectured that if T is a tree, then H λ (T ) = 2 if and only if T = P n , n > 4. We show that this conjecture does not hold by providing a counterexample. Also, we give some classes of trees with maximum number of holes two.

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 fu-fv⩾2 if d(u,v)=1 and fu-fv⩾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 Hλ(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.