The L(2,1)-labelling number of distance graphs G(D), denoted by λ(D), isstudied. It is shown that distance graphs satisfy λ(G) ≤Δ~2. Moreover, we prove λ({1,2, ..., k})=2k +2 and λ({1,3,..., 2k -1}) =2k + 2 for any fixed positive integer k. Suppose k, a ∈ N and k,a≥2. If k≥a, then λ({a, a + 1,..., a + k - 1}) = 2(a + k-1). Otherwise, λ({a, a + 1, ..., a + k- 1}) ≤min{2(a + k-1), 6k -2}. When D consists of two positive integers,6≤λ(D)≤8. For thespecial distance sets D = {k, k + 1}(any k ∈N), the upper bound of λ(D) is improved to 7.
Let G be a simple graph with n (≥5) vertices. In this paper, we prove that if G is 3 connected and satisfies that d(u,v) =2 implies max ｛d(u),d(v)｝ ≥(n+1) /2 for every pair of vertices u and v in G , then for any two vertices x, y of G , there are (x,y) paths of length from 6 to n -1 in G , and there are (x,y) paths of length from 5 to n -1 in G unless G[(x )] = G[(y )]≌ K 4 or K 5 , or G [(x )], G [(y )] are complete and (x)∩(y)=.
A graph is called star extremal if its fractional chromatic number is equal to its circular chromatic number. We first give a necessary and sufficient condition for a graph G to have circular chromatic number V(G)/α(G) (where V(G) is the vertex number of G and α(G) is its independence number). From this result, we get a necessary and sufficient condition for a vertex-transitive graph to be star extremal as well as a necessary and sufficient condition for a circulant graph to be star extremal. Using these conditions, we obtain several classes of star extremal graphs.
L (2, 1)-labeling number, λ(G( Z , D)) , of distance graph G( Z , D) is studied. For general finite distance set D , it is shown that 2D+2≤λ(G( Z , D))≤D 2+3D. Furthermore, λ(G( Z , D)) ≤8 when D consists of two prime positive odd integers is proved. Finally, a new concept to study the upper bounds of λ(G) for some special D is introduced. For these sets, the upper bound is improved to 7.
Let G be a 2 connected graph with n vertices. In this paper, we prove that if there exist two vertices of any there independent vertices in G such that the sum of whose degree is at least n , then G is pancyclic, or G is K n/2,n/2 , or G is K n/2,n/2 -e , or G is a cycle of length 5.
