Blocks in Constrained Random Graphs with Fixed Average Degree

Abstract : This work is devoted to the study of typical properties of random graphs from classes with structural constraints, like for example planar graphs, with the additional restriction that the average degree is fixed. More precisely, within a general analytic framework, we provide sharp concentration results for the number of blocks (maximal biconnected subgraphs) in a random graph from the class in question. Among other results, we discover that essentially such a random graph belongs with high probability to only one of two possible types: it either has blocks of at most logarithmic size, or there is a \emphgiant block that contains linearly many vertices, and all other blocks are significantly smaller. This extends and generalizes the results in the previous work [K. Panagiotou and A. Steger. Proceedings of the 20th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA '09), pp. 432-440, 2009], where similar statements were shown without the restriction on the average degree.
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Krattenthaler, Christian and Strehl, Volker and Kauers, Manuel. 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009), 2009, Hagenberg, Austria. Discrete Mathematics and Theoretical Computer Science, DMTCS Proceedings vol. AK, 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009), pp.733-744, 2009, DMTCS Proceedings
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Konstantinos Panagiotou. Blocks in Constrained Random Graphs with Fixed Average Degree. Krattenthaler, Christian and Strehl, Volker and Kauers, Manuel. 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009), 2009, Hagenberg, Austria. Discrete Mathematics and Theoretical Computer Science, DMTCS Proceedings vol. AK, 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009), pp.733-744, 2009, DMTCS Proceedings. 〈hal-01185402〉

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