scholarly journals Circular Chromatic Number of Signed Graphs

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. 

scholarly journals The Chromatic Number of a Signed Graph

10.37236/4938 ◽  
2016 ◽  
Vol 23 (1) ◽  
Author(s):  
Edita Máčajová ◽  
André Raspaud ◽  
Martin Škoviera
Keyword(s):  
Chromatic Number ◽  
Planar Graphs ◽  
Natural Extension ◽  
Signed Graph ◽  
Substantial Part ◽  
Signed Graphs ◽  
Polynomial Invariants ◽  
Graph Colourings ◽  
Definition Of ◽  
Proper Vertex

In 1982, Zaslavsky introduced the concept of a proper vertex colouring of a signed graph $G$ as a mapping $\phi\colon V(G)\to \mathbb{Z}$ such that for any two adjacent vertices $u$ and $v$ the colour $\phi(u)$ is different from the colour $\sigma(uv)\phi(v)$, where is $\sigma(uv)$ is the sign of the edge $uv$. The substantial part of Zaslavsky's research concentrated on polynomial invariants related to signed graph colourings rather than on the behaviour of colourings of individual signed graphs. We continue the study of signed graph colourings by proposing the definition of a chromatic number for signed graphs which provides a natural extension of the chromatic number of an unsigned graph. We establish the basic properties of this invariant, provide bounds in terms of the chromatic number of the underlying unsigned graph, investigate the chromatic number of signed planar graphs, and prove an extension of the celebrated Brooks' theorem to signed graphs.

scholarly journals Strong Oriented Chromatic Number of Planar Graphs without Short Cycles

2008 ◽  
Vol Vol. 10 no. 1 ◽  
Author(s):  
Mickael Montassier ◽  
Pascal Ochem ◽  
Alexandre Pinlou
Keyword(s):  
Abelian Group ◽  
Chromatic Number ◽  
Planar Graphs ◽  
Oriented Graph ◽  
Minimal Order ◽  
Outerplanar Graphs ◽  
International Audience

International audience Let M be an additive abelian group. An M-strong-oriented coloring of an oriented graph G is a mapping f from V(G) to M such that f(u) <> j(v) whenever uv is an arc in G and f(v)−f(u) <> −(f(t)−f(z)) whenever uv and zt are two arcs in G. The strong oriented chromatic number of an oriented graph is the minimal order of a group M such that G has an M-strong-oriented coloring. This notion was introduced by Nesetril and Raspaud [Ann. Inst. Fourier, 49(3):1037-1056, 1999]. We prove that the strong oriented chromatic number of oriented planar graphs without cycles of lengths 4 to 12 (resp. 4 or 6) is at most 7 (resp. 19). Moreover, for all i ≥ 4, we construct outerplanar graphs without cycles of lengths 4 to i whose oriented chromatic number is 7.

scholarly journals On the k-restricted structure ratio in planar and outerplanar graphs

2008 ◽  
Vol Vol. 10 no. 3 (Graph and Algorithms) ◽  
Author(s):  
Gruia Călinescu ◽  
Cristina G. Fernandes
Keyword(s):  
Approximation Algorithms ◽  
Planar Graphs ◽  
Rates Of Convergence ◽  
Simple Graph ◽  
Maximum Weight ◽  
Outerplanar Graphs ◽  
Weighted Version ◽  
International Audience ◽  
Unweighted Case

Graphs and Algorithms International audience A planar k-restricted structure is a simple graph whose blocks are planar and each has at most k vertices. Planar k-restricted structures are used by approximation algorithms for Maximum Weight Planar Subgraph, which motivates this work. The planar k-restricted ratio is the infimum, over simple planar graphs H, of the ratio of the number of edges in a maximum k-restricted structure subgraph of H to the number edges of H. We prove that, as k tends to infinity, the planar k-restricted ratio tends to 1 = 2. The same result holds for the weighted version. Our results are based on analyzing the analogous ratios for outerplanar and weighted outerplanar graphs. Here both ratios tend to 1 as k goes to infinity, and we provide good estimates of the rates of convergence, showing that they differ in the weighted from the unweighted case.

scholarly journals Circular Chromatic Number of Planar Graphs of Large Odd Girth

10.37236/1569 ◽  
2001 ◽  
Vol 8 (1) ◽  
Author(s):  
Xuding Zhu
Keyword(s):  
Chromatic Number ◽  
Planar Graphs ◽  
Circular Chromatic Number ◽  
Connected Graphs ◽  
Restricted Version

It was conjectured by Jaeger that $4k$-edge connected graphs admit a $(2k+1, k)$-flow. The restriction of this conjecture to planar graphs is equivalent to the statement that planar graphs of girth at least $4k$ have circular chromatic number at most $2+ {{1}\over {k}}$. Even this restricted version of Jaeger's conjecture is largely open. The $k=1$ case is the well-known Grötzsch 3-colour theorem. This paper proves that for $k \geq 2$, planar graphs of odd girth at least $8k-3$ have circular chromatic number at most $2+{{1}\over {k}}$.

scholarly journals Toward a Laplacian spectral determination of signed ∞-graphs

Filomat ◽  
2018 ◽  
Vol 32 (6) ◽  
pp. 2283-2294 ◽  
Author(s):  
Mohammad Iranmanesh ◽  
Mahboubeh Saheli
Keyword(s):  
Simple Graph ◽  
Spectral Characterization ◽  
Signed Graph ◽  
Spectral Determination ◽  
Laplacian Spectrum ◽  
Signed Graphs ◽  
Spectral Invariants

A signed graph consists of a (simple) graph G=(V,E) together with a function ? : E ? {+,-} called signature. Matrices can be associated to signed graphs and the question whether a signed graph is determined by the set of its eigenvalues has gathered the attention of several researchers. In this paper we study the spectral determination with respect to the Laplacian spectrum of signed ?-graphs. After computing some spectral invariants and obtain some constraints on the cospectral mates, we obtain some non isomorphic signed graphs cospectral to signed ?-graphs and we study the spectral characterization of the signed ?-graphs containing a triangle.

scholarly journals On the Aα-Eigenvalues of Signed Graphs

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1990
Author(s):  
Germain Pastén ◽  
Oscar Rojo ◽  
Luis Medina
Keyword(s):  
Adjacency Matrix ◽  
Undirected Graph ◽  
Upper Bounds ◽  
Signed Graph ◽  
Lower And Upper Bounds ◽  
Signed Graphs ◽  
Basic Properties ◽  
Underlying Graph ◽  
Positive Semidefiniteness ◽  
Vertex Degrees

For α∈[0,1], let Aα(Gσ)=αD(G)+(1−α)A(Gσ), where G is a simple undirected graph, D(G) is the diagonal matrix of its vertex degrees and A(Gσ) is the adjacency matrix of the signed graph Gσ whose underlying graph is G. In this paper, basic properties of Aα(Gσ) are obtained, its positive semidefiniteness is studied and some bounds on its eigenvalues are derived—in particular, lower and upper bounds on its largest eigenvalue are obtained.

scholarly journals On the Unitary Cayley Signed Graphs

10.37236/716 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Deepa Sinha ◽  
Pravin Garg
Keyword(s):  
Positive Integer ◽  
Cayley Graph ◽  
Signed Graph ◽  
Signed Graphs ◽  
Prime Factors ◽  
Vertex Set ◽  
Unitary Cayley Graph ◽  
Sigma E

A $signed graph$ (or $sigraph$ in short) is an ordered pair $S = (S^u, \sigma)$, where $S^u$ is a graph $G = (V, E)$ and $\sigma : E\rightarrow \{+,-\}$ is a function from the edge set $E$ of $S^u$ into the set $\{+, -\}$. For a positive integer $n > 1$, the unitary Cayley graph $X_n$ is the graph whose vertex set is $Z_n$, the integers modulo $n$ and if $U_n$ denotes set of all units of the ring $Z_n$, then two vertices $a, b$ are adjacent if and only if $a-b \in U_n$. For a positive integer $n > 1$, the unitary Cayley sigraph $\mathcal{S}_n = (\mathcal{S}^u_n, \sigma)$ is defined as the sigraph, where $\mathcal{S}^u_n$ is the unitary Cayley graph and for an edge $ab$ of $\mathcal{S}_n$, $$\sigma(ab) = \begin{cases} + & \text{if } a \in U_n \text{ or } b \in U_n,\\ - & \text{otherwise.} \end{cases}$$ In this paper, we have obtained a characterization of balanced unitary Cayley sigraphs. Further, we have established a characterization of canonically consistent unitary Cayley sigraphs $\mathcal{S}_n$, where $n$ has at most two distinct odd prime factors.

scholarly journals Знаковые графы эквивалентности степени тоша

2020 ◽  
Author(s):  
R. Rajendra ◽  
P. S. K. Redy
Keyword(s):  
Structural Characterization ◽  
Signed Graph ◽  
Signed Graphs ◽  
Underlying Graph ◽  
Marked Graph ◽  
Multiple Edges ◽  
Sigma E

The Tosha-degree of an edge $\alpha $ in a graph $\Gamma$ without multiple edges, denoted by $T(\alpha)$, is the number of edges adjacent to $\alpha$ in $\Gamma$, with self-loops counted twice. A signed graph (marked graph) is an ordered pair $\Sigma=(\Gamma,\sigma)$ ($\Sigma =(\Gamma, \mu)$), where $\Gamma=(V,E)$ is a graph called the underlying graph of $\Sigma$ and $\sigma : E \rightarrow \{+,-\}$ ($\mu : V \rightarrow \{+,-\}$) is a function. In this paper, we define the Tosha-degree equivalence signed graph of a given signed graph and offer a switching equivalence characterization of signed graphs that are switching equivalent to Tosha-degree equivalence signed graphs and $ k^{th}$ iterated Tosha-degree equivalence signed graphs. It is shown that for any signed graph $\Sigma$, its Tosha-degree equivalence signed graph $T(\Sigma)$ is balanced and we offer a structural characterization of Tosha-degree equivalence signed graphs

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

2014 ◽  
Vol 80 (4) ◽  
pp. 268-276 ◽  
Author(s):  
Jaroslav Nešetřil ◽  
Patrice Ossona de Mendez
Keyword(s):  
Chromatic Number ◽  
Circular Chromatic Number ◽  
Large Girth
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