A condition for Hamiltonicity in Sparse Random Graphs with a Fixed Degree Sequence

TL;DR

This paper establishes a new criterion involving the parameter (G) for determining Hamiltonicity and k-factors in sparse random graphs with a fixed degree sequence, extending understanding of graph properties under degree constraints.

Contribution

It introduces a novel parameter (G) that simplifies the detection of Hamilton cycles and k-factors in sparse random graphs with given degree sequences.

Findings

Hamiltonicity reduces to calculating (G)

Existence of k-factors reduces to calculating (G) for (G)

Results apply to graphs with minimum degree at least 4 and specific tail conditions

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 $\beta_k(G) = \max e(A, B) + k(|A|-|B|) - d(A)$, 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 $\beta_k(G_{n, {\bf d}})$.