Circular chromatic number of signed graphs
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In this paper, a signed graph is defined as a pair (G, σ) where G is a graph and σ : E(G) → {+, −} is a signature which assigns to each edge of G a sign.Abstract:
A signed graph is a pair (G, σ), where G is a graph and σ : E(G) → {+, −} is a signature which assigns to each edge of G a sign. Various notions of coloring of signed graphs have been studied. In this paper, we extend circular coloring of graphs to signed graphs. Given a signed graph (G, σ) a circular r-coloring of (G, σ) is an assignment ψ of points of a circle of circumference r to the vertices of G such that for every edge e = uv of G, if σ(e) = +, then ψ(u) and ψ(v) have distance at least 1, and if σ(e) = −, then ψ(v) and the antipodal of ψ(u) have distance at least 1. The circular chromatic number χ c (G, σ) of a signed graph (G, σ) is the infimum of those r for which (G, σ) admits a circular r-coloring. For a graph G, we define the signed circular chromatic number of G to be max{χ c (G, σ) : σ is a signature of G}. We study basic properties of circular coloring of signed graphs and develop tools for calculating χ c (G, σ). We explore the relation between the circular chromatic number and the signed circular chromatic number of graphs, and present bounds for the signed circular chromatic number of some families of graphs. In particular, we determine the supremum of the signed circular chromatic number of k-chromatic graphs of large girth, of simple bipartite planar graphs, d-degenerate graphs, simple outerplanar graphs and series-parallel graphs. We construct a signed planar simple graph whose circular chromatic number is 4 + 2 3. This is based and improves on a signed graph built by Kardos and Narboni as a counterexample to a conjecture of Macajova, Raspaud, and Skoviera.read more
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Book ChapterDOI
Circular Coloring of Signed Bipartite Planar Graphs
Reza Naserasr,Zhouningxin Wang +1 more
TL;DR: The notion of circular coloring of signed bipartite planar graphs has been studied in this paper, where the main question is to find the smallest even value such that for every signed simple planar graph with negative-girth at least f(f(1) = 4, f(2) = 6 and f(3) = 8, for a positive integer ε > 0.
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Density of $C_{-4}$-critical signed graphs
TL;DR: It is concluded that all signed bipartite planar graphs of negative girth at least 8 map to C −4, showing that 8 is the best possible and disproving a conjecture of Naserasr, Rollova and Sopena, in extension of the 4CT.
Journal ArticleDOI
Symmetric Set Coloring of Signed Graphs
TL;DR: In this article , it was shown that the symset-chromatic number gives the minimum partition of a signed graph into independent sets and non-bipartite antibalanced subgraphs.
Journal ArticleDOI
Mapping sparse signed graphs to (K2k,M) $({K}_{2k},M)$
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Circular (4 − \epsilon)-coloring of some classes of signed graphs
TL;DR: In this paper, an improved upper bound of 4 − 2 n+1 2 for the circular chromatic number of a signed 2-degenerate simple graph on n vertices was given.