Infinitely Many Trees with Maximum Number of Holes Zero, One, and 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 |f(u) − f(v)|⩾2 if d(u, v) = 1 and |f(u) − f(v)|⩾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)