On the 4-color theorem for signed graphs

Kardoš, František; Narboni, Jonathan

doi:10.1016/j.ejc.2020.103215

Cited by 16 publications

(19 citation statements)

References 9 publications

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“…It was conjectured in [16] that every planar simple graph admits a 0-free 4-coloring. However, this conjecture was disproved in [12] using a dual notion. A direct proof of a counterexample is given in [18].…”

Section: Circular Chromatic Number Of Signed Graph Classesmentioning

confidence: 99%

“…The construction is by assembling certain gadgets. Similar to the gadget of [12], we start with a mini-gadget depicted in Figure 5 Note that the minimality of the length implies that the two end points of the interval…”

Section: Circular Chromatic Number Of Signed Graph Classesmentioning

confidence: 99%

“…With our definition of circular chromatic number and development in this work, one may restate a conjecture of [16] as to "circular chromatic number of the class of signed planar simple graphs is 4". However, this conjecture is recently disproved in [12]. The first counterexample provided in [12] is essentially the subgraph K 3 (I) of the signed graph of Theorem 54 (they become a same signed graph after a switching).…”

Section: Signed Planar Graphsmentioning

confidence: 99%

“…However, this conjecture is recently disproved in [12]. The first counterexample provided in [12] is essentially the subgraph K 3 (I) of the signed graph of Theorem 54 (they become a same signed graph after a switching). The work of [12] is based on the dual interpretation of the circular four-coloring of signed planar graphs.…”

Section: Signed Planar Graphsmentioning

confidence: 99%

“…Máčajová, Raspaud, and Škoviera [16] conjectured that every signed planar simple graph is 4-colorable. By Proposition 14, this is equivalent to say that χ s c (G) 4 for every planar graph G. Kardos and Narboni [12] refuted this conjecture by constructing a non-4-colorable signed planar graph. Our construction improves on the example of Kardos and Narboni.…”

Section: Introductionmentioning

confidence: 99%

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Circular Chromatic Number of Signed Graphs

Naserasr

Wang

Zhu

2021

Electron. J. Combin.

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A signed graph is a pair $(G, \sigma)$, where $G$ is a graph (loops and multi edges allowed) and $\sigma: E(G) \to \{+, -\}$ 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 $(G, \sigma)$ with no positive loop, a circular $r$-coloring of $(G, \sigma)$ is an assignment $\psi$ of points of a circle of circumference $r$ to the vertices of $G$ such that for every edge $e=uv$ of $G$, if $\sigma(e)=+$, then $\psi(u)$ and $\psi(v)$ have distance at least $1$, and if $\sigma(e)=-$, then $\psi(v)$ and the antipodal of $\psi(u)$ have distance at least $1$. The circular chromatic number $\chi_c(G, \sigma)$ of a signed graph $(G, \sigma)$ is the infimum of those $r$ for which $(G, \sigma)$ admits a circular $r$-coloring. For a graph $G$, we define the signed circular chromatic number of $G$ to be $\max\{\chi_c(G, \sigma): \sigma \text{ is a signature of $G$}\}$. We study basic properties of circular coloring of signed graphs and develop tools for calculating $\chi_c(G, \sigma)$. 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 $4+\frac{2}{3}$. 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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Section: Circular Chromatic Number Of Signed Graph Classesmentioning

confidence: 99%

Section: Circular Chromatic Number Of Signed Graph Classesmentioning

confidence: 99%

Section: Signed Planar Graphsmentioning

confidence: 99%

Section: Signed Planar Graphsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Circular Chromatic Number of Signed Graphs

Naserasr

Wang

Zhu

2021

Electron. J. Combin.

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Density of 3‐critical signed graphs

Beaudou,

Haxell,

Nurse

et al. 2024

Journal of Graph Theory

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We say that a signed graph is ‐critical if it is not ‐colorable but every one of its proper subgraphs is ‐colorable. Using the definition of colorability due to Naserasr, Wang, and Zhu that extends the notion of circular colorability, we prove that every 3‐critical signed graph on vertices has at least edges, and that this bound is asymptotically tight. It follows that every signed planar or projective‐planar graph of girth at least 6 is (circular) 3‐colorable, and for the projective‐planar case, this girth condition is best possible. To prove our main result, we reformulate it in terms of the existence of a homomorphism to the signed graph , which is the positive triangle augmented with a negative loop on each vertex.

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Circular Coloring of Signed Bipartite Planar Graphs

Naserasr

Wang

2021

Trends in Mathematics

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On the 4-color theorem for signed graphs

Cited by 16 publications

References 9 publications

Circular Chromatic Number of Signed Graphs

Circular Chromatic Number of Signed Graphs

Density of 3‐critical signed graphs

Circular Coloring of Signed Bipartite Planar Graphs

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