Hamilton cycles in highly connected and expanding graphs

Hefetz, Dan; Krivelevich, Michael; Szabó, Tibor

doi:10.1007/s00493-009-2362-0

Cited by 45 publications

(62 citation statements)

References 22 publications

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“…The next lemma is a slight strengthening of Claim 2.2 from [13] with a similar proof. It shows that in any graph having the small and large expansion properties for any path P 0 and its endpoint v 1 many other endpoints can be created by a small number of rotations with fixed endpoint v 1 .…”

Section: Proofmentioning

confidence: 66%

“…We obtain the corresponding sets Z and U = V \ (D ∪ Z) and conclude that the induced graph G[U ] satisfies the small expansion property S(s/2, g/2). Theorem 2.5 from [13] states that for every choice of the expansion parameter r with 12 ≤ r ≤ √ n, every n-vertex graph G satisfying S r, n ln r r ln n and L n ln r 1035 ln n is Hamiltonian. Hence, applying this statement to G[U ] with s = n α , g = 4αn/n α and r = n α /2, we see that G[U ] is Hamiltonian.…”

Section: Proofmentioning

confidence: 99%

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On covering expander graphs by hamilton cycles

Glebov

Krivelevich

Szabó

2012

Random Struct Algorithms

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Abstract. The problem of packing Hamilton cycles in random and pseudorandom graphs has been studied extensively. In this paper, we look at the dual question of covering all edges of a graph by Hamilton cycles and prove that if a graph with maximum degree ∆ satisfies some basic expansion properties and contains a family of (1 − o(1))∆/2 edge disjoint Hamilton cycles, then there also exists a covering of its edges by (1 + o(1))∆/2 Hamilton cycles. This implies that for every α > 0 and every p ≥ n α−1 there exists a covering of all edges of G(n, p) by (1 + o(1))np/2 Hamilton cycles asymptotically almost surely, which is nearly optimal.

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Section: Proofmentioning

confidence: 66%

Section: Proofmentioning

confidence: 99%

On covering expander graphs by hamilton cycles

Glebov

Krivelevich

Szabó

2012

Random Struct Algorithms

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“…We will use the following special case of a theorem from [11]: Then G is hamiltonian, for sufficiently large n.…”

Section: Enforcing a Hamilton Cyclementioning

confidence: 99%

Avoider–Enforcer games

Hefetz

Krivelevich

Szabó

2007

Journal of Combinatorial Theory, Series A

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Let p and q be positive integers and let H be any hypergraph. In a (p, q, H) Avoider-Enforcer game two players, called Avoider and Enforcer, take turns selecting previously unclaimed vertices of H. Avoider selects p vertices per move and Enforcer selects q vertices per move. Avoider loses if he claims all the vertices of some hyperedge of H; otherwise Enforcer loses. We prove a sufficient condition for Avoider to win the (p, q, H) game. We then use this condition to show that Enforcer can win the (1, q) perfect matching game on K 2n for every q cn/ log n for an appropriate constant c, and the (1, q) Hamilton cycle game on K n for every q cn log log log log n/ log n log log log n for an appropriate constant c. We also determine exactly those values of q for which Enforcer can win the (1, q) connectivity game on K n . This result is quite surprising as it substantially differs from its Maker-Breaker analog. Our method extends easily to improve a result of Lu [X. Lu, A note on biased and non-biased games, Discrete Appl. Math. 60 (1995) 285-291], regarding forcing an opponent to pack many pairwise edge disjoint spanning trees in his graph.

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“…a Hamilton path between any given pair of vertices in a random graph. This can be achieved using known tools (say, those from [13]). Here too the constant C 2 = C 2 (α) has to grow as α → 0.…”

Section: Introductionmentioning

confidence: 99%