TY - JOUR

T1 - Compatible Hamilton cycles in Dirac graphs

AU - Krivelevich, Michael

AU - Lee, Choongbum

AU - Sudakov, Benny

N1 - Publisher Copyright:
© 2017, János Bolyai Mathematical Society and Springer-Verlag Berlin Heidelberg.

PY - 2017/8/1

Y1 - 2017/8/1

N2 - 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 = {Fv}v∈V such that for every v∈V, the set Fv is a family of unordered pairs Fv ⊆ {{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 Fv 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′}∉Fv, 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.

AB - 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 = {Fv}v∈V such that for every v∈V, the set Fv is a family of unordered pairs Fv ⊆ {{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 Fv 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′}∉Fv, 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.

UR - http://www.scopus.com/inward/record.url?scp=84988382989&partnerID=8YFLogxK

U2 - 10.1007/s00493-016-3328-7

DO - 10.1007/s00493-016-3328-7

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AN - SCOPUS:84988382989

SN - 0209-9683

VL - 37

SP - 697

EP - 732

JO - Combinatorica

JF - Combinatorica

IS - 4

ER -