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Brahim Benmedjdoub, Isma Bouchemakh and Éric Sopena

Abstract

A 2-distance k-coloring of a graph G is a mapping from V (G) to the set of colors {1,. . ., k} such that every two vertices at distance at most 2 receive distinct colors. The 2-distance chromatic number χ 2(G) of G is then the smallest k for which G admits a 2-distance k-coloring. For any finite set of positive integers D = {d 1, . . ., d}, the integer distance graph G = G(D) is the infinite graph defined by V (G) = ℤ and uvE(G) if and only if |vu| ∈ D. We study the 2-distance chromatic number of integer distance graphs for several types of sets D. In each case, we provide exact values or upper bounds on this parameter and characterize those graphs G(D) with χ2(G(D)) = ∆(G(D)) + 1.

Open access

Robert A. Beeler, Teresa W. Haynes and Kyle Murphy

Abstract

Let G be a graph with vertex set V and a distribution of pebbles on the vertices of V. A pebbling move consists of removing two pebbles from a vertex and placing one pebble on a neighboring vertex, and a rubbling move consists of removing a pebble from each of two neighbors of a vertex v and placing a pebble on v. We seek an initial placement of a minimum total number of pebbles on the vertices in V, so that no vertex receives more than one pebble and for any given vertex vV, it is possible, by a sequence of pebbling and rubbling moves, to move at least one pebble to v. This minimum number of pebbles is the 1-restricted optimal rubbling number. We determine the 1-restricted optimal rubbling numbers for Cartesian products. We also present bounds on the 1-restricted optimal rubbling number.

Open access

Yuval

Studies in Jewish Music

Open access

Nick Bailey, Joanna L. Stewart and Jon Minton

Abstract

The suburbanisation of poverty has been noted in the cities of a large number of countries, including the UK. The main drivers are labour market restructuring on the one hand, and market-driven change in the housing system on the other although social and housing policies are also factors. This paper explores the possible consequences for the welfare of low-income groups in relation to two dimensions: exposure to air pollution and access to good quality schools. Results show that, for these groups, suburbanisation has had mixed impacts on welfare. In most cities, suburbanisation is likely to bring improvements in air quality but there are only a minority where it improves access to good quality schools. Overall, it is clear that suburbanising low income households enjoy fewer of the benefits of suburban locations than middle class households.