Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2109.13627

Generalising the achromatic number to Zaslavsky's colourings of signed graphs

Julien Bensmail·François Dross·Nacim Oijid·Éric Sopena
View original source ↗Read source document ↗Export metadata ↓

Abstract

The chromatic number, which refers to the minimum number of colours required to colour the vertices of graphs properly, is one of the most central notions of the graph chromatic theory. Several of its aspects of interest have been investigated in the literature, including variants for modifications of proper colourings. These variants include, notably, the achromatic number of graphs, which is the maximum number of colours required to colour the vertices of graphs properly so that each possible combination of distinct colours is assigned along some edge. The behaviours of this parameter have led to many investigations of interest, bringing to light both similarities and discrepancies with the chromatic number. This work takes place in a recent trend aiming at extending the chromatic theory of graphs to the realm of signed graphs, and, in particular, at investigating how classic results adapt to the signed context. Most of the works done in that line to date are with respect to two main generalisations of proper colourings of signed graphs, attributed to Zaslavsky and Guenin. Generalising the achromatic number to signed graphs was initiated recently by Lajou, his investigations being related to Guenin's colourings. We here pursue this line of research, but with taking Zaslavsky's colourings as our notion of proper colourings. We study the general behaviour of our resulting variant of the achromatic number, mainly by investigating how known results on the classic achromatic number generalise to our context. Our results cover, notably, bounds, standard operations on graphs, and complexity aspects.

Explore related subjects

cs.DMmath.CO

Keep this discovery

BibTeXRIS

Julien Bensmail, François Dross, Nacim Oijid, Éric Sopena. 2021-09-28. Generalising the achromatic number to Zaslavsky's colourings of signed graphs. https://arxiv.org/abs/2109.13627

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

An FPTAS for Two-Machine Open-Shop Scheduling with a Single Unavailability Interval

We consider the two-machine open-shop scheduling problem in which one machine is unavailable during a fixed interval. We study the resumable setting: an operation interrupted by the unavailability interval may resume, without penalty, when the machine becomes available. The objective is to minimize the makespan. Although the problem is NP-hard and several approximation algorithms are known, whether it admits a fully polynomial-time approximation scheme (FPTAS) has remained open for two decades. We resolve this question affirmatively by giving the first FPTAS, thereby strengthening the previously known polynomial-time approximation scheme (PTAS). As an intermediate result, we develop a new pseudo-polynomial dynamic program with seven state dimensions, improving on the ten-dimensional formulation in the literature.

cs.DM↗
arXiv2026

Generalized Graph Search Trees

Graph search algorithms and their corresponding graph search trees are commonly used in algorithmic graph theory. In recent years, the recognition problem of these graph search trees has received significant attention. So far, the research has focused on two types of search trees: first-in trees that behave like BFS-trees and last-in trees that behave like DFS-trees. The search tree paradigms differ from each other by the parent a vertex is connected to. In first-in trees, it is the first visited neighbor, while in last-in trees it is the last neighbor visited before that vertex. Here, we will generalize these concepts of graph search trees by allowing every preceding neighbor of a vertex to be the parent. We study the complexity of the recognition problem of these generalized graph search trees. We present NP-completeness proofs for most searches. We also show that the problem is trivial for Generic Search and polynomial-time solvable for several searches on bipartite graphs and chordal graphs. We also study the question how fixing the start vertex influences the complexity of the problem.

cs.DM↗
arXiv2026

The exact asymptotic constant in the metric dimension of Jaccard space

Let $X$ be a finite set with $|X|=n$ and let $\mathrm{Jac}(a,b)=|a\,\triangle\, b|/|a\cup b|$ be the Jaccard distance on the power set $2^X$. Lladser and Paradise recently proved that the metric dimension of $(2^X,\mathrm{Jac})$ is $\Theta(n/\ln n)$, with the constant left open; their bounds are $(\ln 2)\,n/\ln n\lesssim \beta(2^X,\mathrm{Jac})\lesssim 2\ln(2e)\,n/\ln n$. We determine the constant: \[ \beta(2^X,\mathrm{Jac})=\frac{2n}{\log_2 n}\,(1+o(1))=(2\ln 2)\,\frac{n}{\ln n}\,(1+o(1)). \] The proof identifies the problem, on each ``slice'' of subsets of fixed cardinality, with the Erd\H{o}s--R\'enyi coin-weighing problem for a spring scale (the problem of \emph{detecting matrices}). The lower bound is the Erd\H{o}s--R\'enyi entropy argument applied to the middle slice; the upper bound follows from the explicit detecting families of Lindstr\"om and of Cantor and Mills, augmented by a single extra landmark that reveals cardinality.

cs.DM↗