Looseness of Planar Graphs

International audienceA face of a vertex coloured plane graph is called loose if the number of colours used on its vertices is at least three. The looseness of a plane graph G is the minimum k such that any surjective k-colouring involves a loose face. In this paper we prove that the looseness of a connected plane graph G equals the maximum number of vertex disjoint cycles in the dual graph G increased by 2. We also show upper bounds on the looseness of graphs based on the number of vertices, the edge connectivity, and the girth of the dual graphs. These bounds improve the result of Negami for the looseness of plane triangulations. We also present infinite classes of graphs where the equalities are attained

The Archaeology of Rock Art in Western Arnhem Land, Australia (Terra Australis 47)

Western Arnhem Land, in the Top End of Australia’s Northern Territory, has a rich archaeological landscape, ethnographic record and body of rock art that displays an astonishing array of imagery on shelter walls and ceilings. While the archaeology goes back to the earliest period of Aboriginal occupation of the continent, the rock art represents some of the richest, most diverse and visually most impressive regional assemblages anywhere in the world. To better understand this multi-dimensional cultural record, The Archaeology of Rock Art in Western Arnhem Land, Australia focuses on the nature and antiquity of the region’s rock art as revealed by archaeological surveys and excavations, and the application of novel analytical methods. This volume also presents new findings by which to rethink how Aboriginal peoples have socially engaged in and with places across western Arnhem Land, from the north to the south, from the plains to the spectacular rocky landscapes of the plateau. The dynamic nature of Arnhem Land rock art is explored and articulated in innovative ways that shed new light on the region’s deep time Aboriginal history