The Chromatic Number of a Signed Graph

Máčajová, Edita; Raspaud, André; Škoviera, Martin

doi:10.37236/4938

Cited by 63 publications

(87 citation statements)

References 12 publications

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“…Recently, Naserasr et al [4] introduced a concept of colorings by means of graph homomorphism. Máčajová et al [3] modified Zaslavsky's approach as follows. If = 2 + 1, then let = {0, ±1, … , ± }, and if = 2 , then let = {±1, … , ± }.…”

mentioning

confidence: 99%

Circular coloring of signed graphs

Kang

Steffen

2017

Journal of Graph Theory

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Let k, d (2d ≤ k) be two positive integers. We generalize the well studied notions of (k, d)-colorings and of the circular chromatic number χ c to signed graphs. This implies a new notion of colorings of signed graphs, and the corresponding chromatic number χ.Some basic facts on circular colorings of signed graphs and on the circular chromatic number are proved, and differences to the results on unsigned graphs are analyzed. In particular, we show that the difference between the circular chromatic number and the chromatic number of a signed graph is at most 1. Indeed, there are signed graphs where the difference is 1. On the other hand, for a signed graph on n vertices, if the difference is smaller than 1, then there exists n > 0, such that the difference is at most 1 − n .We also show that notion of (k, d)-colorings is equivalent to r-colorings (see [10]

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confidence: 99%

Circular coloring of signed graphs

Kang

Steffen

2017

Journal of Graph Theory

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“…It was conjectured by Máčajová, Raspaud andŠkoviera [21] that every planar graph is signed 4-colourable, and conjectured by Kündgen and Ramamurthi [20] Proof. Assume G is a signed Z 4 -colourable graph and L is a {1, 1, 2}-assignment of G. We may assume that colours in the lists are positive integers.…”

Section: Signed Graph Colouring and λ-Choosabilitymentioning

confidence: 99%

“…A signed graph is a pair (G, σ) such that G is a graph and σ ∶ E → {−1, +1} is a signature which assigns to each edge a sign. Colouring of signed graphs was first studied by Zalslavsky [30] in the 1980's and has attracted a lot of recent attention [21,14,15]. A set I of integers is called symmetric if for any integer i, i ∈ I implies that −i ∈ I.…”

Section: Introductionmentioning

confidence: 99%

“…We say a graph G is signed k-colourable (respectively, signed Z k -colourable) if for any signature σ of G, the signed graph (G, σ) is k-colourable (respectively, Z k -colourable). It was conjectured by Máčajová, Raspaud andŠkoviera [21] that every planar graph is signed 4-colourable, and conjectured by Kang and Steffen [16] that every planar graph is signed Z 4 -colourable. An assignment L of a graph G is called symmetric if for each vertex v of G, L(v) is a symmetric set of integers.…”

Section: Introductionmentioning

confidence: 99%

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A refinement of choosability of graphs

Zhu

2020

Journal of Combinatorial Theory, Series B

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Assume k is a positive integer, λ = {k 1 , k 2 , . . . , k q } is a partition of k and G is a graph. A λ-assignment of G is a k-assignment L of G such that the colour setIt follows from the definition that if λ = {k}, then λ-choosability is the same as k-choosability, if λ = {1, 1, . . . , 1}, then λ-choosability is equivalent to k-colourability. For the other partitions of k sandwiched between {k} and {1, 1, . . . , 1} in terms of refinements, λ-choosability reveals a complex hierarchy of colourability of graphs. We prove that for two partitions λ, λ ′ of k, every λchoosable graph is λ ′ -choosable if and only if λ ′ is a refinement of λ. Then we study λ-choosability of special families of graphs. The Four Colour Theorem says that every planar graph is {1, 1, 1, 1}-choosable. A very recent result of Kemnitz and Voigt implies that for any partition λ of 4 other than {1, 1, 1, 1}, there is a planar graph which is not λ-choosable. We observe that, in contrast to the fact that there are non-4-choosable 3-chromatic planar graphs, every 3-chromatic planar graph is {1, 3}-choosable, and that if G is a planar graph whose dual G * has a connected spanning Eulerian subgraph, then G is {2, 2}-choosable. We prove that if n is a positive even integer, λ is a partition of n − 1 in which each part is at most 3, then K n is edge λ-choosable. Finally we study relations between λ-choosability of graphs and colouring of signed graphs and generalized signed graphs. A conjecture of Máčajová, Raspaud andŠkoviera that every planar graph is signed 4-colcourable is recently disproved by Kardoš and Narboni. We prove that every signed 4-colourable graph is weakly 4-choosable, and every signed Z 4colourable graph is {1, 1, 2}-choosable. The later result combined with the above result of Kemnitz and Voigt disproves a conjecture of Kang and Steffen that every planar graph is signed Z 4 -colourable. We shall show that a graph constructed by Wegner in 1973 is also a counterexample to Kang and Steffen's conjecture, and present a new construction of a non-{1, 3}-choosable planar graphs.

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“…The colors +a and −a have the same magnitude, but are opposite. These are the same signed color sets used in both [12] and [5] to study signed vertex coloring. Definition 3.1.…”

Section: Edge Coloringsmentioning

confidence: 99%

Edge coloring signed graphs

Behr

2020

Discrete Mathematics

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We define a method for edge coloring signed graphs and what it means for such a coloring to be proper. Our method has many desirable properties: it specializes to the usual notion of edge coloring when the signed graph is allnegative, it has a natural definition in terms of vertex coloring of a line graph, and the minimum number of colors required for a proper coloring of a signed simple graph is bounded above by ∆ + 1 in parallel with Vizing's Theorem. In fact, Vizing's Theorem is a special case of the more difficult theorem concerning signed graphs.

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The Chromatic Number of a Signed Graph

Cited by 63 publications

References 12 publications

Circular coloring of signed graphs

Circular coloring of signed graphs

A refinement of choosability of graphs

Edge coloring signed graphs

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