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Hamiltonicity in connected regular graphs
In 1980, Jackson proved that every 2-connected -regular graph with at most
vertices is Hamiltonian. This result has been extended in several papers.
In this note, we determine the minimum number of vertices in a connected
-regular graph that is not Hamiltonian, and we also solve the analogous
problem for Hamiltonian paths. Further, we characterize the smallest connected
-regular graphs without a Hamiltonian cycle.Comment: 5 page
On highly regular strongly regular graphs
In this paper we unify several existing regularity conditions for graphs,
including strong regularity, -isoregularity, and the -vertex condition.
We develop an algebraic composition/decomposition theory of regularity
conditions. Using our theoretical results we show that a family of non rank 3
graphs known to satisfy the -vertex condition fulfills an even stronger
condition, -regularity (the notion is defined in the text). Derived from
this family we obtain a new infinite family of non rank strongly regular
graphs satisfying the -vertex condition. This strengthens and generalizes
previous results by Reichard.Comment: 29 page
Regular cross sections of Borel flows
Any free Borel flow is shown to admit a cross section with only two possible
distances between adjacent points. Non smooth flows are proved to be Lebesgue
orbit equivalent if and only if they admit the same number of invariant ergodic
probability measures.Comment: Minor improvements in expositio
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