Dynamic Chromatic Number of Regular Graphs

A dynamic coloring of a graph $G$ is a proper coloring such that for every vertex $v\in V(G)$ of degree at least 2, the neighbors of $v$ receive at least 2 colors. It was conjectured [B. Montgomery. {\em Dynamic coloring of graphs}. PhD thesis, West Virginia University, 2001.] that if $G$ is a $k$-regular graph, then $\chi_2(G)-\chi(G)\leq 2$. In this paper, we prove that if $G$ is a $k$-regular graph with $\chi(G)\geq 4$, then $\chi_2(G)\leq \chi(G)+\alpha(G^2)$. It confirms the conjecture for all regular graph $G$ with diameter at most 2 and $\chi(G)\geq 4$. In fact, it shows that $\chi_2(G)-\chi(G)\leq 1$ provided that $G$ has diameter at most 2 and $\chi(G)\geq 4$. Moreover, we show that for any $k$-regular graph $G$, $\chi_2(G)-\chi(G)\leq 6\ln k+2$. Also, we show that for any $n$ there exists a regular graph $G$ whose chromatic number is $n$ and $\chi_2(G)-\chi(G)\geq 1$. This result gives a negative answer to a conjecture of [A. Ahadi, S. Akbari, A. Dehghan, and M. Ghanbari. \newblock On the difference between chromatic number and dynamic chromatic number of graphs. \newblock {\em Discrete Math.}, In press].Comment: 8 page

Evaluation of differences in S. pneumoniae colonization among children with and without clinically diagnosed asthma/wheeze

Streptococcus pneumoniae colonization is the nasopharynx is common in young children. Colonization of S. pneumoniae is a necessary precursor for invasive pneumococcal disease (IPD), which is a major cause of morbidity and mortality among children less than five years of age globally. Co-morbidities such as asthma have been identified as risk factors for IPD but little is known about why. Children with co-morbidities have a higher likelihood of progressing to IPD because they are colonized at higher rates or because their immune systems respond differently than children without co-morbidities. In addition, vaccination was introduced in 2010 to help decrease pneumococcal colonization rates from the 13 most common serotypes. We used data from the pediatric primary care clinic at Boston Medical Center to study the relationship between asthma/wheeze and S. pneumoniae colonization among children under the age of five years. Information about colonization serotype distribution was also assessed in this study. Data was accessed from 3098 children from 4–59 months old visiting the pediatric primary care clinic at Boston Medical Center from July 2010 to March 2014. In multivariable logistic regression models, the odds of colonization increased 80% (OR 1.80, 95% CI 1.2, 2.6) in children with asthma/wheeze under 24 months old. Adjustment for presence of URTI or recent exposure to antibiotics slightly mitigates this relationship. Children with clinically diagnosed asthma/wheeze have 80% increased odds of being colonized than children without asthma/wheeze