Minimal Oriented Graphs of Diameter 2

Füredi, Zoltán; Horák, Peter; Pareek, C. M.; Zhu, Xuding

doi:10.1007/pl00021182

Cited by 11 publications

(2 citation statements)

References 7 publications

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“…It is known that an oriented graph O is an oclique if and only if each pair of its non-adjacent vertices are connected by a directed 2-path [10]. Also, it is known [6] that the number of arcs of an oclique on n vertices may vary between (n log n − 3n 2 ) and n 2 . However, the oriented chromatic polynomial for any oclique O on n vertices is the same [18]:…”

Section: Counting Homomorphismsmentioning

confidence: 99%

A Homomorphic Polynomial for Oriented Graphs

Das

Ghosh²,

Prabhu³

et al. 2023

Electron. J. Combin.

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In this article, we define a function that counts the number of (onto) homomorphisms of an oriented graph. We show that this function is always a polynomial and establish it as an extension of the notion of chromatic polynomials. We study algebraic properties of this function. In particular we show that the coefficients of these polynomials have the alternating sign property and that the polynomials associated to the independent sets have relations with the Stirling numbers of the second kind.

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Section: Counting Homomorphismsmentioning

confidence: 99%

A Homomorphic Polynomial for Oriented Graphs

Das

Ghosh²,

Prabhu³

et al. 2023

Electron. J. Combin.

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“…A number of works are dedicated in quest of finding the exact description of this function [18,20,23].…”

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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