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- Daphne Der-Fen Liu, Xuding Zhu
- SIAM J. Discrete Math.
- 2005

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) − dG(u, v) + 1. The span of f… (More)

- Daphne Der-Fen Liu
- Discrete Mathematics
- 2008

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)−dG(u, v)+1, where dG(u, v) is the distancee between u and v. We prove a lower bound for the radio… (More)

- Seog-Jin Kim, Young Soo Kwon, Daphne Der-Fen Liu, Xuding Zhu
- Electr. J. Comb.
- 2012

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 K2?(k−1),3 is not on-line chromatic-choosable. Based on this result, the on-line version of Ohba Conjecture is modified as… (More)

- Gerard J. Chang, Daphne Der-Fen Liu, Xuding Zhu
- J. Comb. Theory, Ser. B
- 1999

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)

- Gerard J. Chang, Wen-Tsai Ke, David Kuo, Daphne Der-Fen Liu, Roger K. Yeh
- Discrete Mathematics
- 2000

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 de ned as the minimum m… (More)

- Daphne Der-Fen Liu, Xuding Zhu
- Journal of Graph Theory
- 2004

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 vertex set Z and in which i and j are adjacent whenever |i − j| ∈ M . Supported in part by the National Science Foundation under grant DMS 9805945. Supported in part by the National Science Council, R. O. C., under grant… (More)

- Daphne Der-Fen Liu, Roger K. Yeh
- Ars Comb.
- 1997

- Ko-Wei Lih, Daphne Der-Fen Liu, Xuding Zhu
- SIAM J. Discrete Math.
- 1999

- Gerard J. Chang, Justie Su-tzu Juan, Daphne Der-Fen Liu
- Ars Comb.
- 2001

- Daphne Der-Fen Liu, Melanie Xie
- Ars Comb.
- 2009

Let G be a connected graph. For any two vertices u and v, let d(u, v) denote the distance between u and v in G. The maximum distance between any pair of vertices is called the diameter of G and denoted by diam(G). A radio-labeling (or multi-level distance labeling) with span k for G is a function f that assigns to each vertex with a label from the set {0,… (More)