Graph Classes (Dis)satisfying the Zagreb Indices Inequality

International audience{Recently Hansen and Vukicevic proved that the inequality $M_1/n \leq M_2/m$, where $M_1$ and $M_2$ are the first and second Zagreb indices, holds for chemical graphs, and Vukicevic and Graovac proved that this also holds for trees. In both works is given a distinct counterexample for which this inequality is false in general. Here, we present some classes of graphs with prescribed degrees, that satisfy $M_1/n \leq M_2/m$: Namely every graph $G$ whose degrees of vertices are in the interval $[c; c + \sqrt c]$ for some integer $c$ satisies this inequality. In addition, we prove that for any $\Delta \geq 5$, there is an infinite family of graphs of maximum degree $\Delta$ such that the inequality is false. Moreover, an alternative and slightly shorter proof for trees is presented, as well\ as for unicyclic graphs

A note on the metric and edge metric dimensions of 2-connected graphs

For a given graph $G$, the metric and edge metric dimensions of $G$, $\dim(G)$ and ${\rm edim}(G)$, are the cardinalities of the smallest possible subsets of vertices in $V(G)$ such that they uniquely identify the vertices and the edges of $G$, respectively, by means of distances. It is already known that metric and edge metric dimensions are not in general comparable. Infinite families of graphs with pendant vertices in which the edge metric dimension is smaller than the metric dimension are already known. In this article, we construct a 2-connected graph $G$ such that $\dim(G)=a$ and ${\rm edim}(G)=b$ for every pair of integers $a,b$, where $4\le b<a$. For this we use subdivisions of complete graphs, whose metric dimension is in some cases smaller than the edge metric dimension. Along the way, we present an upper bound for the metric and edge metric dimensions of subdivision graphs under some special conditions.Comment: 12 page

3-facial colouring of plane graphs

International audienceA plane graph is l-facially k-colourable if its vertices can be coloured with k colours such that any two distinct vertices on a facial segment of length at most l are coloured differently. We prove that every plane graph is 3-facially 11-colourable. As a consequence, we derive that every 2-connected plane graph with maximum face-size at most 7 is cyclically 11-colourable. These two bounds are for one off from those that are proposed by the (3l+1)-Conjecture and the Cyclic Conjecture

Common agenda or Europe's agenda? International protection, human rights and migration from the Horn of Africa

This paper examines the relationship between international protection, human rights and migration in the context of EU Agenda on Migration which aims to ‘tackle migration upstream’ and reduce the number of arrivals to Europe from countries in the Horn of Africa (HoA) (Eritrea, Somalia, Ethiopia, South Sudan and Sudan). These initiatives are underpinned by assumptions about the factors driving migration from the region, including the idea that poverty rather than political oppression and human rights abuse is the principal cause. The paper draws on interview and survey data with 128 people originating from HoA countries and arriving in Europe between March 2011 and October 2016 to show that conflict, insecurity and human rights abuse in countries of origin and neighbouring countries drive decisions to move. This evidence challenges the premises underlying the European Agenda. Moreover, a lack of coherence between the EU’s ambitions to control irregular migration and the rights-violating States with which Europe seeks to engage threatens to create further political destabilisation which may ultimately increase, rather than decrease, outward migration from the region. Agreements between the EU and HoA should be re-centred to focus on compliance with international human rights standards rather than States’ willingness and to prevent irregular migration to Europe