TY - JOUR

T1 - A note on circular colorings of edge-weighted digraphs

AU - Lin, Wu-Hsiung

AU - Yeh, Hong Gwa

PY - 2011/1/1

Y1 - 2011/1/1

N2 - An edge-weighted digraph (G, ℓ) is a strict digraph G together with a function ℓ assigning a real weight ℓuv to each arc uv. (G, ℓ) is symmetric if uv is an arc implies that so is vu. A circular r-coloring of (G, ℓ) is a function φ assigning each vertex of G a point on a circle of perimeter r such that, for each arc uv of G, the length of the arc from φ(u) to φ(v) in the clockwise direction is at least ℓuv. The circular chromatic number χc(G, ℓ) of (G, ℓ) is the infimum of real numbers r such that (G, ℓ) has a circular r-coloring. Suppose that (G, ℓ) is an edge-weighted symmetric digraph with positive weights on the arcs. Let T be a {0, 1}-function on the arcs of G with the property that T(uv) + T(vu) = 1 for each arc uv in G. In this note we show that if, for each dicycle C of G satisfying, then (G, ℓ) has a circular r-coloring.

AB - An edge-weighted digraph (G, ℓ) is a strict digraph G together with a function ℓ assigning a real weight ℓuv to each arc uv. (G, ℓ) is symmetric if uv is an arc implies that so is vu. A circular r-coloring of (G, ℓ) is a function φ assigning each vertex of G a point on a circle of perimeter r such that, for each arc uv of G, the length of the arc from φ(u) to φ(v) in the clockwise direction is at least ℓuv. The circular chromatic number χc(G, ℓ) of (G, ℓ) is the infimum of real numbers r such that (G, ℓ) has a circular r-coloring. Suppose that (G, ℓ) is an edge-weighted symmetric digraph with positive weights on the arcs. Let T be a {0, 1}-function on the arcs of G with the property that T(uv) + T(vu) = 1 for each arc uv in G. In this note we show that if, for each dicycle C of G satisfying, then (G, ℓ) has a circular r-coloring.

KW - Circular chromatic number

KW - Digraph

UR - http://www.scopus.com/inward/record.url?scp=80053376936&partnerID=8YFLogxK

U2 - 10.11650/twjm/1500406428

DO - 10.11650/twjm/1500406428

M3 - Article

AN - SCOPUS:80053376936

VL - 15

SP - 2159

EP - 2167

JO - Taiwanese Journal of Mathematics

JF - Taiwanese Journal of Mathematics

SN - 1027-5487

IS - 5

ER -