Density of C−4-critical signed graphs

Naserasr, Reza; Pham, Lan Anh; Wang, Zhouningxin

doi:10.1016/j.jctb.2021.11.002

Cited by 9 publications

(6 citation statements)

References 13 publications

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“…For signed bipartite planar graphs of negative girth at least 8, the upper bound of 8 3 for their circular chromatic numbers is proved in [5]. For signed bipartite planar graphs of negative girth 2k, k ≥ 5, the best current bound follows from recent results of [3].…”

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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“…While to determine whether a graph admits a homomorphism to an odd cycle is an interesting and difficult (NP‐complete) problem [17], it is easy to decide whether graph

G

admits a homomorphism to an even cycle. In contrast, to decide whether a signed graph

\hat{G}

admits a homomorphism to a negative even cycle is difficult and related to many challenging conjectures [18, 20, 22]. The dual concept of admitting homomorphisms to cycles is “modulo

\ell

‐orientation.” We propose the following generalization to signed graphs.…”

Section: Modulo ℓ $\Ell $‐Orientation and Homomorphism To Cyclesmentioning

confidence: 99%

Circular flows in mono‐directed signed graphs

Li,

Naserasr,

Wang

et al. 2024

Journal of Graph Theory

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In this paper, the concept of circular ‐flow in a mono‐directed signed graph is introduced. That is a pair , where is an orientation on and satisfies that for each positive edge and for each negative edge , and the total in‐flow equals the total out‐flow at each vertex. This is the dual notion of circular colorings of signed graphs and is distinct from the concept of circular flows in bi‐directed graphs associated with signed graphs studied in the literature. We first explore the connection between circular ‐flows and modulo ‐orientations in signed graphs. Then we focus on the upper bounds for the circular flow indices of signed graphs in terms of the edge‐connectivity, where the circular flow index of a signed graph is the minimum value such that it admits a circular ‐flow. We prove that every 3‐edge‐connected signed graph admits a circular 6‐flow and every 4‐edge‐connected signed graph admits a circular 4‐flow. More generally, for , we show that every ‐edge‐connected signed graph admits a circular ‐flow, every ‐edge‐connected signed graph has a circular ‐flow with , and every ‐edge‐connected signed graph admits a circular ‐flow. Moreover, the ‐edge‐connectivity condition is shown to be sufficient for a signed Eulerian graph to admit a circular ‐flow, and applying this result to planar graphs, we conclude that every signed bipartite planar graph of negative girth at least admits a homomorphism to the negative even cycles .

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“…In this context, it is natural to ask for each core subgraphs of K M ( , ) there are two notable ones to consider: (1) the negative 4-cycle (we refer to [17] for recent progress on this problem) and (2) K M ( , )…”

Section: The Chromatic Number and Homomorphisms Tomentioning

confidence: 99%

“…In this context, it is natural to ask for each core subgraphs of

({K}_{4,4},M)

which families of signed planar graphs map to it. Of such subgraphs of

({K}_{4,4},M)

there are two notable ones to consider: (1) the negative 4‐cycle (we refer to [17] for recent progress on this problem) and (2)

({K}_{3,3},M)

. Considering Theorem 2.1, the question of mapping signed bipartite planar graphs captures the 3‐coloring problem of planar graphs.…”

Section: Homomorphisms To (Kkkm) $({K}_{kk}m)$ and (K2km) $({K}_{2k}m)$mentioning

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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Density of C−4-critical signed graphs

Cited by 9 publications

References 13 publications

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

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

Circular flows in mono‐directed signed graphs

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

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