Hamiltonicity in connected regular graphs

In 1980, Jackson proved that every 2-connected $k$-regular graph with at most $3k$ vertices is Hamiltonian. This result has been extended in several papers. In this note, we determine the minimum number of vertices in a connected $k$-regular graph that is not Hamiltonian, and we also solve the analogous problem for Hamiltonian paths. Further, we characterize the smallest connected $k$-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, $k$-isoregularity, and the $t$-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 $7$-vertex condition fulfills an even stronger condition, $(3,7)$-regularity (the notion is defined in the text). Derived from this family we obtain a new infinite family of non rank $3$ strongly regular graphs satisfying the $6$-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