A Note on Circular Colorings of Edge-weighted Digraphs
Resource
Taiwanese Journal of Mathematics, 15(5), 2159-2167
Journal
Taiwanese Journal of Mathematics
Pages
2159-2167
Date Issued
2011
Date
2011
Author(s)
Lin, Wu-Hsiung
Yeh, Hong-Gwa
Abstract
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.
Type
journal article
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