A condition for Hamiltonicity in Sparse Random Graphs with a Fixed Degree Sequence
Abstract
We consider the random graph $G_{n, {\bf d}}$ chosen uniformly at random from the set of all graphs with a given sparse degree sequence ${\bf d}$. We assume ${\bf d}$ has minimum degree at least 4, at most a power law tail, and place one more condition on its tail. For $k\ge 2$ define $\beta_k(G) = \max e(A, B) + k(AB)  d(A)$, with the maximum taken over disjoint vertex sets $A, B$. It is shown that the problem of determining if $G_{n, {\bf d}}$ contains a Hamilton cycle reduces to calculating $\beta_2(G_{n, {\bf d}})$. If $k\ge 2$ and $\delta\ge k+2$, the problem of determining if $G_{n, {\bf d}}$ contains a $k$factor reduces to calculating $\beta_k(G_{n, {\bf d}})$.
 Publication:

arXiv eprints
 Pub Date:
 January 2020
 arXiv:
 arXiv:2001.05258
 Bibcode:
 2020arXiv200105258J
 Keywords:

 Mathematics  Combinatorics;
 Mathematics  Probability