In this paper we investigate some basic properties of fractional powers. In this regard, we show that for any rational number 1 ≤ 2r+1and a non-bipartite graph G, we show that G 2r+1 2s+1 < G 2p+1 2q+1 . In the sequel, we introduce an equivalent definition for circular chromatic number of graphs in terms of fractional powers. We also present a sufficient condition for equality of chromatic number and circular chromatic number.

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It was conjectured in [S. Akbari, F. Khaghanpoor, and S. Moazzeni. Colorful paths in vertex coloring of graphs. Preprint] that, if $G$ is a connected graph distinct from $C_7$, then there is a $\chi(G)$-coloring of $G$ in which every vertex $v\in V(G)$ is an initial vertex of a path $P$ with $\chi(G)$ vertices whose colors are different. In [S. Akbari, V. Liaghat, and A. Nikzad. Colorful paths in vertex coloring of graphs. Electron. J. Combin. 18(1): P17, 9pp, 2011] this was proved with $\lfloor\frac{\chi(G)}{2} \rfloor $ vertices instead of $\chi(G)$ vertices. We strengthen this to $\chi(G)-1$ vertices. We also prove that every connected graph with at least one edge has a proper $k$-coloring (for some $k$) such that every vertex of color $i$ has a neighbor of color $i+1$ (mod $k$). $C_5$ shows that $k$ may have to be greater than the chromatic number. However, if the graph is connected, infinite and locally finite, and has finite chromatic number, then the $k$-coloring exists for every $k \geq \chi(G)$. In fact, the $k$-coloring can be chosen such that every vertex is a starting vertex of an infinite path such that the color increases by $1$ (mod $k$) along each edge. The method is based on the circular chromatic number $\chi_c(G)$. In particular, we verify the above conjecture for all connected graphs whose circular chromatic number equals the chromatic number.

An r-dynamic k-coloring of a graph G is a proper vertex k-coloring such that the neighbors of any vertex v receive at least min{r, deg(v)} different colors. The r-dynamic chromatic number of G, χ r (G), is defined as the smallest k such that G admits an r-dynamic k-coloring. In this paper we introduce an upper bound for χ r (G) in terms of r, chromatic number, maximum degree and minimum degree. In 2001, Montgomery [9] conjectured that, for a d-regular graph G, χ 2 (G) − χ(G) ≤ 2. In this regard, for a d-regular graph G, we present two upper bounds for χ 2 (G) − χ(G), one of them, ⌈5.437 log d + 2.721⌉, is an improvement of the bound 14.06 log d + 1, proved by Alishahi (2011) [2]. Also, we give an upper bound for χ 2 (G) in terms of chromatic number, maximum degree and minimum degree.

In this work, a novel drug delivery system was developed for thymol. Thymol is an herbal medicine that has low chemical reactivity. The drug delivery platform was Janus Magnetic Nanoparticle. The carrier's core‐shell structure was Fe3O4 nanoparticles (Fe3O4) as core and thin layer SiO2 (SiN) as a shell. The Fe3O4 nanoparticles were synthesized using the co‐precipitation method. A thin layer of SiO2 was covered on the surface of Fe3O4 nanoparticles. The whole Janus magnetic nanoparticle, Fe3O4@SiN, was functionalized with beta‐cyclodextrin (β‐CD) and used as a drug delivery platform. The binding between thymol and the Fe3O4@SiN/β‐CD was made possible through bipolar‐bipolar and hydrogen bonding. The structure, magnetic behavior, and applicability of Fe3O4@SiN for drug delivery purposes were investigated by scanning electron microscopy, X‐ray diffraction spectroscopy, Fourier‐transform infrared spectroscopy, UV–Vis spectroscopy, vibrating sample magnetometer, and energy‐dispersive X‐ray spectroscopy. Vibrating sample magnetometer (VSM) indicated that the Fe3O4@SiN has a weak ferromagnetic behavior (Hc = 65.748G) at temperature 300 K and a permeability of 35.759 × 10−3 emu gG−1. The scanning electron microscopy (SEM) images showed that the particle size of Fe3O4@SiN is 40 nm. Finally, the Fe3O4@SiN/β‐CD was successfully applied to simulated body fluid.