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Editors contains: "Billington, Elizabeth J"

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  1. ; (, Australasian Journal of Combinatorics)
    Albert, Michael; Billington, Elizabeth J (Ed.)
    A k-dispersed labelling of a graph G on n vertices is a labelling of the vertices of G by the integers 1,...,n such that d(i,i+1) ≥ k for 1 ≤ i ≤ n − 1. Here DL(G) denotes the maximum value of k such that G has a k-dispersed labelling. In this paper, we study upper and lower bounds on DL(G). Computing DL(G) is NP-hard. However, we determine the exact value of DL(G) for cycles, paths, grids, hypercubes and complete binary trees. We also give a product construction and we prove a degree-based bound. 
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    Accepted Manuscript Full Text Available
  2. ; (, The Australasian journal of combinatorics)
    Albert, Michael; Billington, Elizabeth J (Ed.)
    In the first partial result toward Steinberg’s now-disproved three coloring conjecture, Abbott and Zhou used a counting argument to show that every planar graph without cycles of lengths 4 through 11 is 3-colorable. Implicit in their proof is a fact about plane graphs: in any plane graph of minimum degree 3, if no two triangles share an edge, then triangles make up strictly fewer than 2/3 of the faces. We show how this result, combined with Kostochka and Yancey’s resolution of Ore’s conjecture for k = 4, implies that every planar graph without cycles of lengths 4 through 8 is 3-colorable. 
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    Accepted Manuscript Full Text Available