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Complete minors and independence number
 SIAM J. Discrete Math
"... Abstract Let G be a graph with n vertices and independence number α. Hadwiger's conjecture implies that G contains a clique minor of order at least n/α. In 1982, Duchet and Meyniel proved that this bound holds within a factor 2. Our main result gives the first improvement on their bound by an ..."
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Abstract Let G be a graph with n vertices and independence number α. Hadwiger's conjecture implies that G contains a clique minor of order at least n/α. In 1982, Duchet and Meyniel proved that this bound holds within a factor 2. Our main result gives the first improvement on their bound by an absolute constant factor. We show that G contains a clique minor of order larger than .504n/α. We also prove related results giving lower bounds on the order of the largest clique minor.
Contributions to the Theory of Colourings, Graph Minors, and Independent Sets
, 2011
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Clique minors in clawfree graphs
 J. Combin. Theory Ser. B
"... Hadwiger’s conjecture states that every graph with chromatic number χ has a clique minor of size χ. Let G be a graph on n vertices with chromatic number χ and stability number α. Then since χα ≥ n, Hadwiger’s conjecture implies that G has a clique minor of size n α. In this paper we prove that this ..."
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Hadwiger’s conjecture states that every graph with chromatic number χ has a clique minor of size χ. Let G be a graph on n vertices with chromatic number χ and stability number α. Then since χα ≥ n, Hadwiger’s conjecture implies that G has a clique minor of size n α. In this paper we prove that this is true for connected clawfree graphs with α ≥ 3. We also show that this result is tight by providing an infinite family of clawfree graphs with α ≥ 3 that do not have a clique minor of size larger than n α
EXTREMAL GRAPH THEORY: RAMSEYTURÁN NUMBERS, CHROMATIC THRESHOLDS, AND MINORS
, 2011
"... This dissertation investigates several questions in extremal graph theory and the theory of graph minors. It consists of three independent parts; the first two parts focus on questions motivated by Turán’s Theorem and the third part investigates a problem related to Hadwiger’s Conjecture. Let H be a ..."
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This dissertation investigates several questions in extremal graph theory and the theory of graph minors. It consists of three independent parts; the first two parts focus on questions motivated by Turán’s Theorem and the third part investigates a problem related to Hadwiger’s Conjecture. Let H be a graph, t an integer, and f(n) a function. Erdős, Hajnal, Sós, and Szemerédi defined the tRamseyTurán number of H, RTt(n, H, f(n)), to be the maximum number of edges in an nvertex, Hfree graph with Ktindependence number less than f(n), where the Ktindependence number of a graph G is the maximum number of vertices in a Ktfree induced graph of G. In the first part of this thesis, we study the RamseyTurán numbers for several graphs and hypergraphs, proving two conjectures of Erdős, Hajnal, Simonovits, Sós, and Szemerédi. In joint work with József Balogh, our first main theorem is to provide the first lower bounds of order Ω(n 2) on RTt(n, Kt+2, o(n)). Our second main theorem is to prove lower bounds on RT(n, TK s (r), o(n)), where TK s (r) is the runiform hypergraph formed from Ks by adding r − 2 new vertices to every edge. Let F be a family of runiform hypergraphs. Introduced by Erdős and Simonovits, the chromatic threshold of F is the infimum of the values c ≥ 0 such that the subfamily of F consisting of hypergraphs with minimum degree at least c ()