Relative clique number of planar signed graphs

Das, Sandip; Ghosh, Prantar; Prabhu, Swathy; Sen, Sagnik

doi:10.1016/j.dam.2018.03.005

Cited by 5 publications

(4 citation statements)

References 18 publications

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“…The notions of absolute and relative cliques exists, and are relevant, for signed graphs also. The following are some works on them: [6,11,10,31].…”

Section: The Significance Of Relative Cliquesmentioning

confidence: 99%

On Relative Clique Number of Triangle-Free Planar Colored Mixed Graphs

Nandi¹,

Sen

Taruni

2022

Lecture Notes in Computer Science

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An (n, m)-graph is a graph with n types of arcs and m types of edges. A homomorphism of an (n, m)-graph G to another (n, m)-graph H is a vertex mapping that preserves the adjacencies along with their types and directions. The order of a smallest (with respect to the number of vertices) such H is the (n, m)-chromatic number of G. Moreover, an (n, m)-relative clique R of an (n, m)-graph G is a vertex subset of G for which no two distinct vertices of R get identified under any homomorphism of G. The (n, m)-relative clique number of G, denoted by ω r(n,m) (G), is the maximum |R| such that R is an (n, m)-relative clique of G. In practice, (n, m)relative cliques are often used for establishing lower bounds of (n, m)-chromatic number of graph families.Generalizing an open problem posed by Sopena [Discrete Mathematics 2016] in his latest survey on oriented coloring, Chakroborty, Das, Nandi, Roy and Sen [Discrete Applied Mathematics 2022] conjectured that ω r(n,m) (G) ≤ 2(2n + m) 2 + 2 for any triangle-free planar (n, m)-graph G and that this bound is tight for all (n, m) = (0, 1). In this article, we positively settle this conjecture by improving the previous upper bound of ω r(n,m) (G) ≤ 14(2n+m) 2 +2 to ω r(n,m) (G) ≤ 2(2n+m) 2 +2, and by finding examples of triangle-free planar graphs that achieve this bound. As a consequence of the tightness proof, we also establish a new lower bound of 2(2n + m) 2 + 2 for the (n, m)-chromatic number for the family of triangle-free planar graphs.

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“…The notions of absolute and relative cliques exists, and are relevant, for signed graphs also. The following are some works on them: [6,11,10,31].…”

Section: The Significance Of Relative Cliquesmentioning

confidence: 99%

On Relative Clique Number of Triangle-Free Planar Colored Mixed Graphs

Nandi¹,

Sen

Taruni

2022

Lecture Notes in Computer Science

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“…In [34] it is shown that the problem of determining the relative or absolute S-clique number of a signed graph is NP-hard. However, in [7] the relative S -clique number is determined for graphs of some families of planar and outerplanar signed graphs.…”

Section: Signed Cliquesmentioning

confidence: 99%

Concepts of signed graph coloring

Steffen,

Vogel

2019

Preprint

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The paper surveys some concepts of signed graph coloring. Introduction and definitionsThis paper surveys concepts of colorings of signed graphs. The majority of them are natural extension/generalizations of vertex coloring and the chromatic number of unsigned graphs. However, it turns out that there are coloring concepts which are equivalent for unsigned graphs but they are not equivalent for signed graphs in general. Consequently, there are several versions of a chromatic number of a signed graph. We give a brief overview of the various concept and relate some of them to each other. We use standard terminology of graph theory. However, we will first give some necessary definitions, some of which are less standard.We consider finite graphs G with vertex set V(G) and edge set E(G). An edge e consists of two half-edges each of which is incident to a vertex. If the half-edges of e are incident to vertices v and w, then the half-edges are denoted by h e (v) and h e (w). If v = w, then e is a loop. We say that e is incident to v and w or that v and w are the end vertices of e and we will also use the term vw to denote e.Let G be a graph and C be a set. A mapping c :The chromatic number of G is the minimum k for which there is a k-coloring of G.Let H(G) be the set of half-edges of G, and H G (v) be the set of half-edges which are incident to v. The cardinality of H G (v) is the degree of v which is denoted by d G (v).The maximum degree of a vertex of G is denoted by ∆(G), and the minimum degree of a vertex of G is denoted by δbe a set of vertices. The subgraph of G induced by X is denoted by G[X], and the set of edges with precisely one end in X is denoted by ∂(X). A circuit is a connected

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“…Over the passed few years, their work has generated increasing attention to the topic, see e.g. [2,6,8,14,16,19]. One reason behind this interest lies in the fact that homomorphisms of signed graphs stand as a natural way for generalizing a number of classical results and conjectures from graph theory, including, especially, ones related to graph minor theory (such as the Four-Color Theorem and Hadwiger's Conjecture).…”

Section: Introductionmentioning

confidence: 99%

On the signed chromatic number of some classes of graphs

Bensmail¹,

Das²,

Nandi³

et al. 2020

Preprint

Self Cite

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A signed graph (G, σ) is a graph G along with a function σ : E(G) → {+, −}. A closed walk of a signed graph is positive (resp., negative) if it has an even (resp., odd) number of negative edges, counting repetitions. A homomorphism of a (simple) signed graph to another signed graph is a vertex-mapping that preserves adjacencies and signs of closed walks. The signed chromatic number of a signed graph (G, σ) is the minimum number of vertices |V (H)| of a signed graph (H, π) to which (G, σ) admits a homomorphism.Homomorphisms of signed graphs have been attracting growing attention in the last decades, especially due to their strong connections to the theories of graph coloring and graph minors. These homomorphisms have been particularly studied through the scope of the signed chromatic number. In this work, we provide new results and bounds on the signed chromatic number of several families of signed graphs (planar graphs, triangle-free planar graphs, K n -minor-free graphs, and bounded-degree graphs).

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Relative clique number of planar signed graphs

Cited by 5 publications

References 18 publications

On Relative Clique Number of Triangle-Free Planar Colored Mixed Graphs

On Relative Clique Number of Triangle-Free Planar Colored Mixed Graphs

Concepts of signed graph coloring

On the signed chromatic number of some classes of graphs

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