A Note on Circular Chromatic Number of Graphs with Large Girth and Similar Problems

Jaroslav Nešetřil; Patrice Ossona de Mendez

doi:10.1002/jgt.21849

Circular Chromatic Number of Signed Graphs

The Electronic Journal of Combinatorics ◽

10.37236/9938 ◽

2021 ◽

Vol 28 (2) ◽

Author(s):

Reza Naserasr ◽

Zhouningxin Wang ◽

Xuding Zhu

Keyword(s):

Chromatic Number ◽

Planar Graphs ◽

Simple Graph ◽

Signed Graph ◽

Signed Graphs ◽

Outerplanar Graphs ◽

Circular Chromatic Number ◽

Basic Properties ◽

Large Girth ◽

Sigma E

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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The circular chromatic number of series-parallel graphs with large girth

Journal of Graph Theory ◽

10.1002/(sici)1097-0118(200004)33:4<185::aid-jgt1>3.0.co;2-n ◽

2000 ◽

Vol 33 (4) ◽

pp. 185-198 ◽

Cited By ~ 5

Author(s):

Chihyun Chien ◽

Xuding Zhu

Keyword(s):

Chromatic Number ◽

Circular Chromatic Number ◽

Large Girth

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A Construction of Uniquely $n$-Colorable Digraphs with Arbitrarily Large Digirth

The Electronic Journal of Combinatorics ◽

10.37236/4823 ◽

2017 ◽

Vol 24 (2) ◽

Author(s):

Michael Severino

Keyword(s):

Chromatic Number ◽

Graph Coloring ◽

Independent Set ◽

Directed Cycle ◽

Circular Chromatic Number ◽

Graph Theoretic ◽

Theoretic Concept ◽

Vertex Set ◽

Large Girth

A natural digraph analogue of the graph-theoretic concept of an `independent set' is that of an `acyclic set', namely a set of vertices not spanning a directed cycle. Hence a digraph analogue of a graph coloring is a decomposition of the vertex set into acyclic sets and we say a digraph is uniquely $n$-colorable when this decomposition is unique up to relabeling. It was shown probabilistically in [A. Harutyunyan et al., Uniquely $D$-colorable digraphs with large girth, Canad. J. Math., 64(6): 1310-1328, 2012] that there exist uniquely $n$-colorable digraphs with arbitrarily large girth. Here we give a construction of such digraphs and prove that they have circular chromatic number $n$. The graph-theoretic notion of `homomorphism' also gives rise to a digraph analogue. An acyclic homomorphism from a digraph $D$ to a digraph $H$ is a mapping $\varphi: V(D) \rightarrow V(H)$ such that $uv \in A(D)$ implies that either $\varphi(u)\varphi(v) \in A(H)$ or $\varphi(u)=\varphi(v)$, and all the `fibers' $\varphi^{-1}(v)$, for $v \in V(H)$, of $\varphi$ are acyclic. In this language, a core is a digraph $D$ for which there does not exist an acyclic homomorphism from $D$ to a proper subdigraph of itself. Here we prove some basic results about digraph cores and construct highly chromatic cores without short cycles.

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Uniquely Colourable Graphs with Large Girth

Canadian Journal of Mathematics ◽

10.4153/cjm-1976-133-5 ◽

1976 ◽

Vol 28 (6) ◽

pp. 1340-1344 ◽

Cited By ~ 24

Author(s):

Béla Bollobás ◽

Norbert Sauer

Keyword(s):

Chromatic Number ◽

Large Girth ◽

Large Chromatic Number

Tutte [1], writing under a pseudonym, was the first to prove that a graph with a large chromatic number need not contain a triangle. The result was rediscovered by Zykov [5] and Mycielski [4]. Erdös [2] proved the much stronger result that for every k ≧ 2 and g there exist a k-chromatic graph whose girth is at least g.

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