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Graph Theory @ Georgia Tech
Graph Theory @ Georgia Tech
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ItemGraphs of Small Rank-Width are Pivot-Minors of Graphs of Small Tree-Width(Georgia Institute of Technology, 2012-05) Oum, Sang-ilWe prove that every graph of rank-width k is a pivot-minor of a graph of tree-width at most 2k. We also prove that graphs of rank-width at most 1, also known as distance-hereditary graphs, are exactly vertex-minors of trees, and graphs of linear rank-width at most 1 are precisely vertex-minors of paths. This is a joint work with O-Joung Kwon.
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ItemBraces and Pfaffian Orientations(Georgia Institute of Technology, 2012-05) Whalen, PeterRobertson, Seymour, Thomas and simultaneously McCuaig answered several equivalent questions. Specifically, when can some of the 1's of a 0-1 square matrix, A, be changed to -1's so that the permanent of A equals the determinant of the modified matrix? When is a hypergraph with n vertices and n hyperedges minimally nonbipartite? When does a bipartite graph have a Pfaffian orientation? Given a digraph, does it have an even directed circuit? When is a square matrix sign non-singular? We provide a much shorter proof using elementary methods for their theorem. This is joint work with Robin Thomas.
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ItemA Refinement of the Triangle Version of the Corradi-Hajnal Theorem(Georgia Institute of Technology, 2012-05) Kostochka, Alexandr V.An important part of the Corradi-Hajnal Theorem says that if n = 3k, then every n-vertex graph G with minimum degree at least 2k=2n/3 contains k vertex-disjoint triangles. The restriction on the minimum degree is sharp. This is equivalent (by switching to the complement) to the statement that every n-vertex graph H with maximum degree at most k-1 has an equitable k-coloring, that is a proper coloring of vertices of H with k colors such that the sizes of color classes differ by at most 1. In 1970, Hajnal and Szemeredi generailzed this result by proving the conjecture of Erdos that every graph with maximum degree at most r has an equitable r+1-coloring. In this talk, we prove a Brooks-type result describing for r\geq n/4 all n-vertex graphs with maximum degree at most r that do not admit an equitable r-coloring. In particular, we conclude that for k\geq 6, at most one 3k-vertex graph with minimum degree at least 2k-1 and independence number at most k does not contain k vertex-disjoint triangles. This is joint work with H. A. Kierstead.
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ItemEmbeddability of Infinite Graphs(Georgia Institute of Technology, 2012-05) Salazar, Gelasio
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ItemOn Matchings and Covers(Georgia Institute of Technology, 2012-05) Haxell, PennyLet H be a hypergraph. A matching in H is a set of pairwise disjoint edges of H. A cover of H is a set C of vertices that meets all edges of H. We discuss a number of conjectures and results that bound the minimum size of a cover of H in terms of the maximum size of a matching in H, for various classes of hypergraph H. For example, we consider r-partite hypergraphs and triangle hypergraphs.
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ItemExtending Polynomials in Combinatorics(Georgia Institute of Technology, 2012-05) Loebl, MartinTutte polynomial is an extension of the chromatic polynomial. Fruitful source of extensions is q-commutation and determinant: q-MacMahon Master theorem is an important example. I will describe the basic setting and show application in other parts of combinatorics.
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ItemClique Immersion in Digraphs(Georgia Institute of Technology, 2012-05) McDonald, JessicaImmersion is a containment relation between graphs (or digraphs) which is defined similarly to the more familiar notion of minors, but is incomparable to it. In this talk we focus on immersing a bidirected clique \vec{K}_t in an Eulerian digraph. We show that every Eulerian digraph with minimum degree at least t(t-1) immerses \vec{K_t}, and for t = 4 this requirement can be lowered to t-1. We also provide a rough structure theorem for \vec{K}_t-immersion in Eulerian digraphs. Joint work with Matt DeVos, Bojan Mohar and Diego Scheide.
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ItemColoring Graphs on Surfaces(Georgia Institute of Technology, 2012-05) Dvorak, ZdenekWe give an overview of the recent progress on coloring embedded graphs.
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ItemOn Tractability of Minimum 0-Extension Problems(Georgia Institute of Technology, 2012-05) Hirai, HiroshiGiven an undirected graph G, a set V including V(G), and a nonnegative cost function c on the set of all pairs on V, The minimum 0-extension problem for G,V,c is to find a 0-extension d of d_G with \sum_xy c(xy)d(x,y) minimum, where a 0-extension is a metric d on V extending d_G such that for every x in V there exists s in V(G) with d(x,s) = 0. This problem, introduced by A. Karzanov in 1998, includes minimum s-t cut problem for G = K_2 and, more generally, multiway cut problem for G = K_n. Our interest is a question raised by Karzanov: What is the graph G for which the minimum 0-extension problem on (fixed) G is polynomially solvable? In this talk, we describe some aspects and new results to this problem.
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ItemPlanarity and Dimension for Graphs and Posets(Georgia Institute of Technology, 2012-05) Trotter, William T.There is a rich history of research relating planarity for graphs and diagrams with the dimension of posets, starting with the elegant characterization of planarity for posets with a zero and a one: they are planar if and only if they have dimension at most 2. Planar posets with a zero (or a one) have dimension at most 3, but Kelly showed that there are planar posets of arbitrarily large dimension. Subsequently, Schnyder proved that a graph is planar if and only if the dimension of its incidence poset is at most 3. Quite recently, Felsner, Wiechert and Trotter have shown that the dimension of a poset with a planar comparability graph is at most 4, while Streib and Trotter have shown that the dimension of a poset with a planar cover graph is bounded as a function of its height.