A note on collections of graphs with non-surjective lambda labelings

The λ-number of a graph G, denoted λ(G), is the smallest integer k such that there exists a function from V(G) into {0,1,2,…,k} under which adjacent vertices receive integers which differ by at least 2 and vertices at distance two receive integers which differ by at least 1. We establish the infinitude of the collection of connected graphs G with fixed maximum degree Δ⩾4 and fixed λ-number Δ+t, 1⩽t⩽Δ-1 such that no λ-labeling of G into {0,1,2,…,λ(G)} is surjective. Also, from among graphs with no surjective λ-labelings, we construct connected graphs with maximum degree 3, λ-number 5 and arbitrarily large order.