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Given a graph G and a positive integer d, an L(d; 1)-labeling of G is a function f that assigns to each vertex of G a non-negative integer such that if two vertices u and v are adjacent, then |f(u) − f(v)|¿d; if u and v are not adjacent but there is a two-edge path between them, then |f(u) − f(v)|¿1. The L(d; 1)-number of G, d (G), is deÿned as the minimum(More)
For a graph G, let diam(G) denote the diameter of G. For any two vertices u and v in G, let d(u, v) denote the distance between u and v. A multi-level distance labeling (or distance labeling) for G is a function f that assigns to each vertex of G a non-negative integer such that for any vertices u and v, |f (u) − f (v)| ≥ diam(G) − d G (u, v) + 1. The span(More)
Let Z be the set of all integers and M a set of positive integers. The distance graph G(Z, M) generated by M is the graph with ver-tex set Z and in which i and j are adjacent whenever |i − j| ∈ M. 1 We investigate the fractional chromatic number and the circular chromatic number of the distance graphs G(Z, M) with clique size at least |M |. We first give a(More)
Given positive integers m,) = (m + sk + 1)/(s + 1) otherwise. The latter result provides a good lower bound for χ(Z, D m,k,s). A general upper bound for χ(Z, D m,k,s) is found. We prove the upper bound can be improved to (m + sk + 1)/(s + 1) + 1 for some values of m, k and s. In particular, when s + 1 is prime, χ(Z, D m,k,s) is either (m + sk + 1)/(s + 1)(More)
Let G be a connected graph with diameter diam(G). The radio number for G, denoted by rn(G), is the smallest integer k such that there exists a function f : V (G) → {0, 1, 2, · · · , k} with the following satisfied for all vertices u and v: |f (u)−f (v)| ≥ diam(G)−d G (u, v)+1, where d G (u, v) is the distancee between u and v. We prove a lower bound for the(More)
For a graph G, we denote its diameter by diam(G), and denote the distance between any two vertices, u and v, by d G (u, v). A multi-level distance labeling (or radio labeling) of G is a function f that assigns to each vertex a non-negative integer such that for any pair of vertices u and v, it is satisfied that |f (u) − f (v)| ≥ diam(G) − d G (u, v) + 1.(More)
The Ohba Conjecture says that every graph G with |V (G)| ≤ 2χ(G) + 1 is chromatic choosable. This paper studies an on-line version of Ohba Conjecture. We prove that unlike the off-line case, for k ≥ 3, the complete multipartite graph K 2(k−1),3 is not on-line chromatic-choosable. Based on this result, the on-line version of Ohba Conjecture is modified as(More)
We discuss relationships among T-colorings of graphs and chromatic numbers, fractional chromatic numbers, and circular chromatic numbers of distance graphs. We first prove that for any finite integral set T that contains 0, the asymptotic T-coloring ratio R(T) is equal to the fractional chromatic number of the distance graph G(Z, D), where D=T&[0]. This(More)