Dirac's theorem for random regular graphs

We prove a `resilience' version of Dirac's theorem in the setting of random regular graphs. More precisely, we show that, whenever $d$ is sufficiently large compared to $\varepsilon>0$, a.a.s. the following holds: let $G'$ be any subgraph of the random $n$-vertex $d$-regular graph $G_{n,d}$ with minimum degree at least $(1/2+\varepsilon)d$. Then $G'$ is Hamiltonian. This proves a conjecture of Ben-Shimon, Krivelevich and Sudakov. Our result is best possible: firstly, the condition that $d$ is large cannot be omitted, and secondly, the minimum degree bound cannot be improved.Comment: Final accepted version, to appear in Combinatorics, Probability & Computin

Cycle factors and renewal theory

For which values of $k$ does a uniformly chosen $3$-regular graph $G$ on $n$ vertices typically contain $n/k$ vertex-disjoint $k$-cycles (a $k$-cycle factor)? To date, this has been answered for $k=n$ and for $k \ll \log n$; the former, the Hamiltonicity problem, was finally answered in the affirmative by Robinson and Wormald in 1992, while the answer in the latter case is negative since with high probability most vertices do not lie on $k$-cycles. Here we settle the problem completely: the threshold for a $k$-cycle factor in $G$ as above is $\kappa_0 \log_2 n$ with $\kappa_0=[1-\frac12\log_2 3]^{-1}\approx 4.82$. Precisely, we prove a 2-point concentration result: if $k \geq \kappa_0 \log_2(2n/e)$ divides $n$ then $G$ contains a $k$-cycle factor w.h.p., whereas if $k<\kappa_0\log_2(2n/e)-\frac{\log^2 n}n$ then w.h.p. it does not. As a byproduct, we confirm the "Comb Conjecture," an old problem concerning the embedding of certain spanning trees in the random graph $G(n,p)$. The proof follows the small subgraph conditioning framework, but the associated second moment analysis here is far more delicate than in any earlier use of this method and involves several novel features, among them a sharp estimate for tail probabilities in renewal processes without replacement which may be of independent interest.Comment: 45 page

Minimum-weight combinatorial structures under random cost-constraints

Recall that Janson showed that if the edges of the complete graph Kn are assigned exponentially distributed independent random weights, then the expected length of a shortest path between a fixed pair of vertices is asymptotically equal to (log n)/n. We consider analogous problems where edges have not only a random length but also a random cost, and we are interested in the length of the minimumlength structure whose total cost is less than some cost budget. For several classes of structures, we determine the correct minimum length structure as a function of the cost-budget, up to constant factors. Moreover, we achieve this even in the more general setting where the distribution of weights and costs are arbitrary, so long as the density f(x) as x → 0 behaves like cxγ for some γ ≥ 0; previously, this case was not understood even in the absence of cost constraints. We also handle the case where each edge has several independent costs associated to it, and we must simultaneously satisfy budgets on each cost. In this case, we show that the minimum-length structure obtainable is essentially controlled by the product of the cost thresholds

Gender-based violence and its association with mental health among Somali women in a Kenyan refugee camp: a latent class analysis

background In conflict-affected settings, women and girls are vulnerable to gender-based violence (GBV). GBV is associated with poor long-term mental health such as anxiety, depression and post-traumatic stress disorder (PTSD). Understanding the interaction between current violence and past conflict-related violence with ongoing mental health is essential for improving mental health service provision in refugee camps. Methods Using data collected from 209 women attending GBV case management centres in the Dadaab refugee camps, Kenya, we grouped women by recent experience of GBV using latent class analysis and modelled the relationship between the groups and symptomatic scores for anxiety, depression and PTSD using linear regression. Results Women with past-year experience of intimate partner violence alone may have a higher risk of depression than women with past-year experience of non-partner violence alone (Coef. 1.68, 95% CI 0.25 to 3.11). Conflict-related violence was an important risk factor for poor mental health among women who accessed GBV services, despite time since occurrence (average time in camp was 11.5 years) and even for those with a past-year experience of GBV (Anxiety: 3.48, 1.85–5.10; Depression: 2.26, 0.51–4.02; PTSD: 6.83, 4.21–9.44). Conclusion Refugee women who experienced past-year intimate partner violence or conflict-related violence may be at increased risk of depression, anxiety or PTSD. Service providers should be aware that compared to the general refugee population, women who have experienced violence may require additional psychological support and recognise the enduring impact of violence that occurred before, during and after periods of conflict and tailor outreach and treatment services accordingly