A refinement of choosability of graphs

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.

Section: Signed Planar Graphsmentioning

confidence: 99%

Circular Chromatic Number of Signed Graphs

Naserasr

Wang

2021

Electron. J. Combin.

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“…We say

G

λ

‐ choosable if

G

L

‐colourable for any

λ

‐list assignment

L

G

. The concept of

λ

‐ choosability was introduced in [14] as a refinement of choosability of graphs. If

∣ λ ∣ = 1

, that is,

λ = {k_{λ}}

, then

λ

‐choosability is the same as

k_{λ}

‐choosability; if

∣ λ ∣ = k_{λ}

, that is,

λ

consists of

k_{λ}

copies of 1, then

λ

‐choosability is the equivalent to

k_{λ}

‐colourable.…”

Section: Introductionmentioning

confidence: 99%

“…It follows from the definitions that if

λ \leq λ'

, then every

λ

‐choosable graph is

λ'

‐choosable. Conversely, it was proved in [14] that if

λ \leq̸ λ'

, then there is a

λ

‐choosable graph which is not

λ'

‐choosable.…”

Section: Introductionmentioning

confidence: 99%

Chromatic λ‐choosable and λ‐paintable graphs

Journal of Graph Theory

2021

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Let ϕ ( k ) be the minimum number of vertices in a non‐ k‐choosable k‐chromatic graph. The Ohba conjecture, confirmed by Noel, Reed and Wu, asserts that ϕ ( k ) ≥ 2 k + 2. This bound is tight if k is even. If k is odd, then it is known that ϕ ( k ) ≤ 2 k + 3 and it is conjectured by Noel that ϕ ( k ) = 2 k + 3. For a multiset λ = { k 1 , k 2 , … , k q } of positive integers, let k λ = ∑ i = 1 q k i. For positive integer a, let m λ ( a ) be the multiplicity of a in λ, and let m λ ( odd ) be the number of elements in λ that are odd integers. A λ‐list assignment of G is a list assignment L such that the colour set ∪ v ∈ V ( G ) L ( v ) can be partitioned into the disjoint union C 1 ∪ C 2 ∪ … ∪ C q of q sets so that for each i and each vertex v of G, ∣ L ( v ) ∩ C i ∣ ≥ k i. We say G is λ‐choosable if G is L‐colourable for any λ‐list assignment L of G. We say λ is trivial if λ consists of k λ copies of 1. It is easy to see that if λ is trivial, then λ‐choosable is equivalent to k λ‐colourable. For any nontrivial λ, let ϕ ( λ ) be the minimum number of vertices in a non‐ λ‐choosable k λ‐chromatic graph. We prove that for any nontrivial λ, 2 k λ + m λ ( 1 ) + 2 ⩽ ϕ ( λ ) ⩽ 2 k λ + m λ ( odd ) + 2. In particular, if m λ ( 1 ) = m λ ( odd ) = t, that is, λ contains no odd integer greater than 1, then ϕ ( λ ) = 2 k λ + t + 2. We also prove that ϕ ( λ ) ⩽ 2 k λ + 5 m λ ( 1 ) + 3. In particular, if m λ ( 1 ) = 0, then 2 k λ + 2 ⩽ ϕ ( λ ) ⩽ 2 k λ + 3. We then introduce the concept of λ‐paintability of graphs, which is an online version of λ‐choosability. Let ψ ( λ ) be the minimum number of vertices in a non‐ λ‐paintable k λ‐chromatic graph. We determine the value of ψ ( λ ) when each integer in λ is at most 2.

“…This bound is best possible, as proved by Voigt [14], who constructed the first non-4-choosable planar graph. Other examples of planar graphs with ch(G) = 5 can be found in [3,11,18].…”

Section: Introductionmentioning

confidence: 99%

The Alon-Tarsi number of a planar graph minus a matching

Grytczuk

Journal of Combinatorial Theory, Series B

2020

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This paper proves that every planar graph G contains a matching M such that the Alon-Tarsi number of G − M is at most 4. As a consequence, G − M is 4paintable, and hence G itself is 1-defective 4-paintable. This improves a result of Cushing and Kierstead [Planar Graphs are 1-relaxed, 4-choosable, European Journal of Combinatorics 31(2010),1385-1397], who proved that every planar graph is 1-defective 4-choosable.