## Abstract

A graph is Hamiltonian if it contains a cycle passing through every vertex exactly once. A celebrated theorem of Dirac from 1952 asserts that every graph on *n*≥3 vertices with minimum degree at least *n*/2 is Hamiltonian. We refer to such graphs as Dirac graphs. In this paper we obtain the following strengthening of this result. Given a graph *G*=(*V*,*E*), an *incompatibility system F* over *G* is a family *F* = {*F*
_{
v
}}_{
v∈V
} such that for every *v*∈*V*, the set *F*
_{
v
} is a family of unordered pairs *F*
_{
v
} ⊆ {{*e*,*e*′}}: *e* ≠ *e*′ ∈ *E*,*e* ∩ *e*′ = {*v*}}. An incompatibility system is *Δ*-*bounded* if for every vertex *v* and an edge *e* incident to *v*, there are at most *Δ* pairs in *F*
_{
v
} containing *e*. We say that a cycle *C* in *G* is *compatible* with *F* if every pair of incident edges *e*,*e*′ of *C* satisfies {*e*,*e*′}∉*F*
_{
v
}, where *v*=*e* ∩ *e*′. This notion is partly motivated by a concept of transition systems defined by Kotzig in 1968, and can be viewed as a quantitative measure of robustness of graph properties. We prove that there is a constant *μ*>0 such that for every *μn*-bounded incompatibility system *F* over a Dirac graph *G*, there exists a Hamilton cycle compatible with *F*. This settles in a very strong form a conjecture of Häggkvist from 1988.

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## References

- [1]
N. Alon and G. Gutin: Properly colored Hamilton cycles in edge-colored complete graphs,

*Random Struct. Algor.***11**(1997), 179–186. - [2]
B. Bollobás and P. Erdős: Alternating hamiltonian cycles,

*Israel J. Math.***23**(1976), 126–131. - [3]
J. Bondy: Paths and cycles, in:

*Handbook of Combinatorics*,*Vol. 1*(edited by R. Graham, M. Grotschel, and L. Lovász), Elsevier, Amsterdam (1995), 5–110. - [4]
J. Böttcher, M. Schacht and A. Taraz: Proof of the bandwidth conjecture of Bollobás and Komlós,

*Mathematische Annalen***343**(2009), 175–205. - [5]
C. Chen and D. Daykin: Graphs with Hamiltonian cycles having adjacent lines different colors,

*J. Combin. Theory B***21**(1976), 135–139. - [6]
D. Christofides, D. Kühn and D. Osthus: Edge-disjoint Hamilton cycles in graphs,

*J. Combin. Theory B***102**(2012), 1035–1060. - [7]
B. Csaba, D. Kühn, A. Lo, D. Osthus and A. Treglown: Proof of the 1-factorization and Hamilton decomposition conjectures,

*Mem. Amer. Math. Soc.*, to appear. - [8]
B. Cuckler and J. Kahn: Hamiltonian cycles in Dirac graphs,

*Combinatorica***29**(2009), 299–326. - [9]
D. E. Daykin: Graphs with cycles having adjacent lines different colors,

*J. Combin. Theory B***20**(1976), 149–152. - [10]
A. Ferber, M. Krivelevich and B.Sudakov: Counting and packing Hamilton cycles in dense graphs and oriented graph,

*J. Combin. Theory B*, to appear. - [11]
P. Heinig: On prisms, Möbius ladders and the cycle space of dense graphs,

*European J. Combin.***36**(2014), 503–530. - [12]
J. Komlós, G. Sárközy and E. Szemerédi: On the Pósa-Seymour conjecture,

*J. Graph Theory***29**(1998), 167–176. - [13]
A. Kotzig: Moves without forbidden transitions in a graph,

*Matematicky časopis***18**(1968), 76–80. - [14]
M. Krivelevich, C. Lee and B. Sudakov: Robust Hamiltonicity of Dirac graphs,

*Trans. Amer. Math. Soc.***366**(2014), 3095–3130. - [15]
M. Krivelevich, C. Lee and B. Sudakov: Compatible Hamilton cycles in random graphs,

*Random Struct. Algor.*, to appear. - [16]
D. Kühn and D. Osthus: Hamilton decompositions of regular expanders: a proof of Kelly's conjecture for large tournaments,

*Adv. Math.***237**(2013), 62–146. - [17]
A. Lo: Properly coloured Hamiltonian cycles in edge-coloured complete graphs,

*Combinatorica*, to appear. - [18]
L. Lovász:

*Combinatorial problems and exercises*, 2nd ed., American Mathematical Soc., Providence (2007). - [19]
L. Pósa: Hamiltonian circuits in random graphs,

*Discrete Math.***14**(1976), 359–364. - [20]
J. Shearer: A property of the colored complete graph,

*Discrete Mathematics***25**(1979), 175–178. - [21]
B. Sudakov and V. Vu: Local resilience of graphs,

*Random Struct. Algor.***33**(2008), 409–433. - [22]
D. Woodall: A sufficient condition for Hamiltonian circuits,

*J. Combin. Theory B***25**(1978), 184–186.

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## Additional information

Research supported in part by USA-Israel BSF Grant 2010115 and by grant 912/12 from the Israel Science Foundation.

Research supported in part by NSF Grant DMS-1362326.

Research supported in part by SNSF grant 200021-149111 and by a USA-Israel BSF grant.

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Krivelevich, M., Lee, C. & Sudakov, B. Compatible Hamilton cycles in Dirac graphs.
*Combinatorica* **37, **697–732 (2017). https://doi.org/10.1007/s00493-016-3328-7

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### Mathematics Subject Classification (2010)

- 05C45