Circular Coloring of Signed Bipartite Planar Graphs

Naserasr, Reza; Wang, Zhouningxin

doi:10.1007/978-3-030-83823-2_33

Cited by 4 publications

(7 citation statements)

References 16 publications

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“…That every signed bipartite planar graph of negative girth at least 6 admits a circular 3coloring is recently proved in [9], noting that this proof uses the 4CT and some extensions of it. On the other hand, the best example of signed bipartite planar graph of negative girth 6 we know has circular chromatic number 14 5 .…”

Section: Discussion and Questionsmentioning

confidence: 99%

Circular $(4-ε)$-coloring of some classes of signed graphs

Kardoš,

Narboni,

Naserasr

et al. 2021

Preprint

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A circular r-coloring of a signed graph (G, σ) is an assignment φ of points of a circle C r of circumference r to the vertices of (G, σ) such that for each positive edge uv of (G, σ) the distance of φ(v) and φ(v) is at least 1 and for each negative edge uv the distance of φ(u) from the antipodal of φ(v) is at least 1. The circular chromatic number of (G, σ), denoted χ c (G, σ), is the infimum of r such that (G, σ) admits a circular r-coloring.This notion is recently defined by Naserasr, Wang, and Zhu who, among other results, proved that for any signed d-degenerate simple graph Ĝ we have χ c ( Ĝ) ≤ 2d. For d ≥ 3, examples of signed d-degenerate simple graphs of circular chromatic number 2d are provided. But for d = 2 only examples of signed 2-degenerate simple graphs of circular chromatic number close enough to 4 are given, noting that these examples are also signed bipartite planar graphs. In this work we first observe the following restatement of the 4-color theorem: If (G, σ) is a signed bipartite planar simple graph where vertices of one part are all of degree 2, then χ c (G, σ) ≤ 16 5 . Motivated by this observation, we provide 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 and an improved upper bound of 4 − 4 ⌊ n+2 2 ⌋for the circular chromatic number of a signed bipartite planar simple graph on n vertices. We then show that each of the bounds is tight for any value of n ≥ 4.

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Section: Discussion and Questionsmentioning

confidence: 99%

Circular $(4-ε)$-coloring of some classes of signed graphs

Kardoš,

Narboni,

Naserasr

et al. 2021

Preprint

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“…Zhang (see [11] and [24]) to be related to a conjecture of Jaeger in the theory of circular flow. A bipartite analogue of Jaeger-Zhang conjecture was introduced in [17] and studied in [3], a first case of which is disproved in [16]. Thus we rather pose the following question:…”

Section: Questions and Remarksmentioning

confidence: 95%

“…That f (3) = 5 is a restatement of the Grötzsch theorem. That f (4) = 8 is proved in [16]. For integers l ≥ 5 it is known that f (l) exists and is finite.…”

Section: Questions and Remarksmentioning

confidence: 99%

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Circular chromatic number of signed graphs

Naserasr

Wang

Zhu

2020

Preprint

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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 Máčajová, Raspaud, and Škoviera.

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“…The importance of the study of the circular chromatic number of signed bipartite planar graphs, especially in relation with the 4-color theorem, is presented in [13]. On the other hand, when restricted to the class of signed bipartite graphs, circular bipartite cliques are determined in [21]. A special case, which has been the focus of this work can be stated as follows: The condition of G σ ( , ) being bipartite in this corollary can in fact be dropped.…”

Section: Connection To Circular Colorings Of Signed Graphsmentioning

confidence: 99%

Mapping sparse signed graphs to (K2k,M) $({K}_{2k},M)$

Naserasr

Škrekovski

Wang

et al. 2023

Journal of Graph Theory

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A homomorphism of a signed graph to is a mapping of vertices and edges of to (respectively) vertices and edges of such that adjacencies, incidences, and the signs of closed walks are preserved. We observe in this work that, for , the ‐coloring problem of a given graph can be captured by homomorphism to from a signed bipartite graph that is built from . Here assigns a negative sign to the edges of a perfect matching and a positive sign to the rest. Motivated by these reformulations and in connection with results on 3‐colorings of planar graphs, such as Grötzsch's theorem, we prove that any signed graph with the maximum average degree strictly less than admits a homomorphism to . For , we show that the maximum average degree being strictly less than 3 would suffice for a signed graph to admit a homomorphism to . Both of these bounds are tight. We discuss applications of our work to signed planar graphs and its connection to the study of homomorphisms of 2‐edge‐colored graphs. Among a number of interesting questions that are left open, a notable one is a possible extension of Steinberg's conjecture for the class of signed bipartite planar graphs.

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Circular Coloring of Signed Bipartite Planar Graphs

Cited by 4 publications

References 16 publications

Circular $(4-ε)$-coloring of some classes of signed graphs

Circular $(4-ε)$-coloring of some classes of signed graphs

Circular chromatic number of signed graphs

Mapping sparse signed graphs to (K2k,M) $({K}_{2k},M)$

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