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Thickness-Two Graphs Part Two: More New Nine-Critical Graphs, Independence Ratio, Cloned Planar Graphs, and Singly and Doubly Outerplanar Graphs

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Abstract

There are numerous means for measuring the closeness to planarity of a graph such as crossing number, splitting number, and a variety of thickness parameters. We focus on the classical concept of the thickness of a graph, and we add to earlier work in [4]. In particular, we offer new 9-critical thickness-two graphs on 17, 25, and 33 vertices, all of which provide counterexamples to a conjecture on independence ratio of Albertson; we investigate three classes of graphs, namely singly and doubly outerplanar graphs, and cloned planar graphs. We give a sharp upper bound for the largest chromatic number of the cloned planar graphs, and we give upper and lower bounds for the largest chromatic number of the former two classes.

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Authors and Affiliations

  1. Department of Computer Science, Department of Mathematics, University of Colorado Denver, Campus Box 109, Denver, CO, 80217, USA

    Ellen Gethner

  2. Department of Physics, Indiana University, 727 E. Third Street, Bloomington, IN, 47405-7105, USA

    Thom Sulanke

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  1. Ellen Gethner
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  2. Thom Sulanke
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Correspondence to Ellen Gethner.

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Gethner, E., Sulanke, T. Thickness-Two Graphs Part Two: More New Nine-Critical Graphs, Independence Ratio, Cloned Planar Graphs, and Singly and Doubly Outerplanar Graphs. Graphs and Combinatorics 25, 197–217 (2009). https://doi.org/10.1007/s00373-008-0833-5

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  • DOI: https://doi.org/10.1007/s00373-008-0833-5

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