A Semiexact Degree Condition for Hamilton Cycles in Digraphs

Demetres Christofides; Peter Keevash; Daniela Kühn; Deryk Osthus

doi:10.1137/090761756

Degree and neighborhood intersection conditions restricted to induced subgraphs ensuring Hamiltonicity of graphs

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830914500438 ◽

2014 ◽

Vol 06 (03) ◽

pp. 1450043

Author(s):

Bo Ning ◽

Shenggui Zhang ◽

Bing Chen

Keyword(s):

Hamilton Cycles ◽

Induced Subgraphs ◽

Degree Sum ◽

Degree Condition

Let claw be the graph K1,3. A graph G on n ≥ 3 vertices is called o-heavy if each induced claw of G has a pair of end-vertices with degree sum at least n, and called 1-heavy if at least one end-vertex of each induced claw of G has degree at least n/2. In this note, we show that every 2-connected o-heavy or 3-connected 1-heavy graph is Hamiltonian if we restrict Fan-type degree condition or neighborhood intersection condition to certain pairs of vertices in some small induced subgraphs of the graph. Our results improve or extend previous results of Broersma et al., Chen et al., Fan, Goodman and Hedetniemi, Gould and Jacobson, and Shi on the existence of Hamilton cycles in graphs.

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An exact minimum degree condition for Hamilton cycles in oriented graphs

Journal of the London Mathematical Society ◽

10.1112/jlms/jdn065 ◽

2008 ◽

Vol 79 (1) ◽

pp. 144-166 ◽

Cited By ~ 20

Author(s):

Peter Keevash ◽

Daniela Kühn ◽

Deryk Osthus

Keyword(s):

Minimum Degree ◽

Hamilton Cycles ◽

Oriented Graphs ◽

Degree Condition

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Resilient Degree Sequences with respect to Hamilton Cycles and Matchings in Random Graphs

The Electronic Journal of Combinatorics ◽

10.37236/8279 ◽

2019 ◽

Vol 26 (4) ◽

Cited By ~ 1

Author(s):

Padraig Condon ◽

Alberto Espuny Díaz ◽

Daniela Kühn ◽

Deryk Osthus ◽

Jaehoon Kim

Keyword(s):

Random Graphs ◽

Perfect Matching ◽

Degree Sequence ◽

Hamilton Cycle ◽

Vertex Degree ◽

Degree Sequences ◽

Hamilton Cycles ◽

Degree Condition ◽

Degree Conditions

Pósa's theorem states that any graph $G$ whose degree sequence $d_1 \le \cdots \le d_n$ satisfies $d_i \ge i+1$ for all $i < n/2$ has a Hamilton cycle. This degree condition is best possible. We show that a similar result holds for suitable subgraphs $G$ of random graphs, i.e.~we prove a `resilience version' of Pósa's theorem: if $pn \ge C \log n$ and the $i$-th vertex degree (ordered increasingly) of $G \subseteq G_{n,p}$ is at least $(i+o(n))p$ for all $i<n/2$, then $G$ has a Hamilton cycle. This is essentially best possible and strengthens a resilience version of Dirac's theorem obtained by Lee and Sudakov. Chvátal's theorem generalises Pósa's theorem and characterises all degree sequences which ensure the existence of a Hamilton cycle. We show that a natural guess for a resilience version of Chvátal's theorem fails to be true. We formulate a conjecture which would repair this guess, and show that the corresponding degree conditions ensure the existence of a perfect matching in any subgraph of $G_{n,p}$ which satisfies these conditions. This provides an asymptotic characterisation of all degree sequences which resiliently guarantee the existence of a perfect matching.

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Hamilton cycles in almost distance-hereditary graphs

Open Mathematics ◽

10.1515/math-2016-0003 ◽

2016 ◽

Vol 14 (1) ◽

pp. 19-28

Author(s):

Bing Chen ◽

Bo Ning

Keyword(s):

Hamiltonian Cycle ◽

Previous Theorem ◽

Hamilton Cycles ◽

Induced Subgraph ◽

Degree Sum ◽

Degree Condition

AbstractLet G be a graph on n ≥ 3 vertices. A graph G is almost distance-hereditary if each connected induced subgraph H of G has the property dH(x, y) ≤ dG(x, y) + 1 for any pair of vertices x, y ∈ V(H). Adopting the terminology introduced by Broersma et al. and Čada, a graph G is called 1-heavy if at least one of the end vertices of each induced subgraph of G isomorphic to K1,3 (a claw) has degree at least n/2, and is called claw-heavy if each claw of G has a pair of end vertices with degree sum at least n. In this paper we prove the following two theorems: (1) Every 2-connected, claw-heavy and almost distance-hereditary graph is Hamiltonian. (2) Every 3-connected, 1-heavy and almost distance-hereditary graph is Hamiltonian. The first result improves a previous theorem of Feng and Guo [J.-F. Feng and Y.-B. Guo, Hamiltonian cycle in almost distance-hereditary graphs with degree condition restricted to claws, Optimazation57 (2008), no. 1, 135–141]. For the second result, its connectedness condition is sharp since Feng and Guo constructed a 2-connected 1-heavy graph which is almost distance-hereditary but not Hamiltonian.

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Counting Hamilton cycles in Dirac hypergraphs

Combinatorics Probability Computing ◽

10.1017/s0963548320000619 ◽

2020 ◽

pp. 1-23

Author(s):

Stefan Glock ◽

Stephen Gould ◽

Felix Joos ◽

Daniela Kühn ◽

Deryk Osthus

Keyword(s):

Hamilton Cycle ◽

Uniform Hypergraph ◽

Similar Estimate ◽

Hamilton Cycles

Abstract A tight Hamilton cycle in a k-uniform hypergraph (k-graph) G is a cyclic ordering of the vertices of G such that every set of k consecutive vertices in the ordering forms an edge. Rödl, Ruciński and Szemerédi proved that for $k\ge 3$ , every k-graph on n vertices with minimum codegree at least $n/2+o(n)$ contains a tight Hamilton cycle. We show that the number of tight Hamilton cycles in such k-graphs is ${\exp(n\ln n-\Theta(n))}$ . As a corollary, we obtain a similar estimate on the number of Hamilton ${\ell}$ -cycles in such k-graphs for all ${\ell\in\{0,\ldots,k-1\}}$ , which makes progress on a question of Ferber, Krivelevich and Sudakov.

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