Hamiltonicity and Consecutive L ( 2 , 1 ) – labelings ∗

For a given graph G of order n, an L(2, 1)-labelling is defined as a function f : V (G) → {0, 1, 2, · · ·} such that |f(u)− f(v)| ≥ 2 when dG(u, v) = 1 and |f(u)− f(v)| ≥ 1 when dG(u, v) = 2, is the minimum length of a path between u and v. A k−L(2, 1)-labelling is an L(2, 1)-labelling such that no label is greater than k. The L(2, 1)-labelling number of G, denoted by λ(G), is the smallest number k such that G has a k-L(2, 1)-labelling. The consecutive L(2, 1)-labelling is a variation of L(2, 1)-labelling under the condition that the integers used are consecutive. The consecutive L(2, 1)-labelling number of G is denoted by λ(G). Obviously, λ(G) ≤ λ(G) ≤ n − 1 if G admits a consecutive L(2, 1)-labelling. In this paper, we consider the graphs with λ(G) = n − 1. The main results include: (1) For any two integers n,m with n ≥ 3 and n− 2 ≤ m ≤ (n−1)(n−2) 2 , there exists a simple graph G of order n and size m with λ(G) = n−1. And the graphs G with λ(G) = n−1 and size n−2 (or size (n−1)(n−2) 2 ) are completely determined; (2) For any two integers n,m with n ≥ 4 and 1 ≤ m ≤ n+1 2 , there exists a simple graph G of order n and C(G) = m with λ(G) = n − 1, where C(G) denote the number of components of G. And the graphs G with λ(G) = n− 1 and C(G) = bn+1 2 c are completely determined; (3) Let G be a connected graph of order n ≥ 6 and its diameter d. If λ(G) = n − 1, then 2 ≤ d ≤ b2 c + 1. Moreover for every two integers n and m with n ≥ 6 and 2 6 m 6 bn+2 2 c, there exists a simple graph G of order n and diameter m with λ(G) = n− 1. (4) For any integers m, n with 2 ≤ m ≤ n− 1, there exists a simple graph G of order n with λ(G) = m.