Two-Page Book Embeddings of 4-Planar Graphs

Back in the Eighties, Heath showed that every 3-planar graph is subhamiltonian and asked whether this result can be extended to a class of graphs of degree greater than three. In this paper we affirmatively answer this question for the class of 4-planar graphs. Our contribution consists of two algorithms: The first one is limited to triconnected graphs, but runs in linear time and uses existing methods for computing hamiltonian cycles in planar graphs. The second one, which solves the general case of the problem, is a quadratic-time algorithm based on the book-embedding viewpoint of the problem.Comment: 21 pages, 16 Figures. A shorter version is to appear at STACS 201

Income-related inequality in obesity and its determinants in Spain: What happens beyond the obesity threshold?

This paper computes and decomposes income-related inequalities in three metrics of obesity, namely, status, depth and severity, for Spain, a European country characterized by a universal health care system with very high and rising obesity prevalence rates. Furthermore, this paper investigates the main determinants of the reduction in obesity inequalities observed over time among the female Spanish population. To compute these inequality indexes, we use cross-sectional and individual-level data gathered from the Spanish National Health Survey. We document income-related inequalities in obesity, that are more pronounced in depth and severity and are to the detriment of poor women in Spain. University education is the most important determinant for all three inequality indexes. We further report that inequalities in obesity tend to decline over time for women, which is explained mainly by a substantial decrease in the degree of inequality in secondary education and a large decrease in the income elasticity of obesity

Liminal Consumption within Nigerian wedding rituals: The interplay between bridal identity and Liminal Gatekeepers

Fagbola, L., Raftopoulou, C., & McEachern, M., Liminal consumption within Nigerian wedding rituals: The interplay between bridal identity and liminal gatekeepers, Marketing Theory (Journal Volume Number and Issue Number) pp. xx-xx. Copyright © [2023] (The Authors). Reprinted by permission of SAGE Publications.This article combines the theoretical lenses of bridal identity and liminal consumption to illustrate the processes of problem-solving, negotiation and reconciliation through which the bride creates her bridal identity, in the Global South context of Nigeria. Most wedding ritual studies typically emphasise the processes of creating and negotiating a successful bridal identity, but few acknowledge the possibilities of failure and its effect upon the liminars. In addition, within liminal consumption studies, the role of liminars’ mentors is often under-theorised. Thus, we contribute to the field by expanding on the concept of ‘liminal gatekeepers’ as the individuals and institutions who control and enforce certain norms associated with the liminal experience. Following an interpretivist approach, the article also advances our understanding of the ways in which the demands of liminal gatekeepers affect the liminars’ experiences and identifies three novel bridal identity outcomes, namely: i) Embedded Bridal Identity; ii) Synthesised Bridal Identity; and iii) Marginalisation. In this way, we advance marketing research around how a liminal consumer identity such as bridal identity is co-constructed between liminars and gatekeepers

On Optimal 2- and 3-Planar Graphs

A graph is $k$-planar if it can be drawn in the plane such that no edge is crossed more than $k$ times. While for $k=1$, optimal $1$-planar graphs, i.e., those with $n$ vertices and exactly $4n-8$ edges, have been completely characterized, this has not been the case for $k \geq 2$. For $k=2,3$ and $4$, upper bounds on the edge density have been developed for the case of simple graphs by Pach and T\'oth, Pach et al. and Ackerman, which have been used to improve the well-known "Crossing Lemma". Recently, we proved that these bounds also apply to non-simple $2$- and $3$-planar graphs without homotopic parallel edges and self-loops. In this paper, we completely characterize optimal $2$- and $3$-planar graphs, i.e., those that achieve the aforementioned upper bounds. We prove that they have a remarkably simple regular structure, although they might be non-simple. The new characterization allows us to develop notable insights concerning new inclusion relationships with other graph classes