Vertex-Coloring 2-Edge-Weighting of Graphs

A $k$-{\it edge-weighting} $w$ of a graph $G$ is an assignment of an integer weight, $w(e)\in \{1,\dots, k\}$, to each edge $e$. An edge weighting naturally induces a vertex coloring $c$ by defining $c(u)=\sum_{u\sim e} w(e)$ for every $u \in V(G)$. A $k$-edge-weighting of a graph $G$ is \emph{vertex-coloring} if the induced coloring $c$ is proper, i.e., $c(u) \neq c(v)$ for any edge $uv \in E(G)$. Given a graph $G$ and a vertex coloring $c_0$, does there exist an edge-weighting such that the induced vertex coloring is $c_0$? We investigate this problem by considering edge-weightings defined on an abelian group. It was proved that every 3-colorable graph admits a vertex-coloring $3$-edge-weighting \cite{KLT}. Does every 2-colorable graph (i.e., bipartite graphs) admit a vertex-coloring 2-edge-weighting? We obtain several simple sufficient conditions for graphs to be vertex-coloring 2-edge-weighting. In particular, we show that 3-connected bipartite graphs admit vertex-coloring 2-edge-weighting

On the Existence of General Factors in Regular Graphs

Let $G$ be a graph, and $H\colon V(G)\to 2^\mathbb{N}$ a set function associated with $G$. A spanning subgraph $F$ of $G$ is called an $H$-factor if the degree of any vertex $v$ in $F$ belongs to the set $H(v)$. This paper contains two results on the existence of $H$-factors in regular graphs. First, we construct an $r$-regular graph without some given $H^*$-factor. In particular, this gives a negative answer to a problem recently posed by Akbari and Kano. Second, by using Lov\'asz's characterization theorem on the existence of $(g, f)$-factors, we find a sharp condition for the existence of general $H$-factors in $\{r, r+1\}$-graphs, in terms of the maximum and minimum of $H$. The result reduces to Thomassen's theorem for the case that $H(v)$ consists of the same two consecutive integers for all vertices $v$, and to Tutte's theorem if the graph is regular in addition.Comment: 10 page

Surface Characterization of Chemically Modified Wood: Dynamic Wettability1

Dynamic wettability of chemically modified yellow-poplar veneer was investigated with sessile water droplets in this study. Dynamic contact angle, decay ratio, spreading ratio, and their changing rates (the wetting slope and K value) were used to illustrate the dynamic wetting process. Dynamic contact angle (α) and droplet height decay ratio (DRh) followed the first order exponential decay equation, whereas the droplet base-diameter spreading ratio (SRφ) fitted the Boltzmann sigmoid model. Wetting behavior of Epolene G-3015 [a maleated polypropylene (MAPP) copolymer with a high molecular weight]-treated wood surface was independent of the retention and wetting time. The retention effect on wetting slopes of >, DRh, and SRφ on poly(ethylene and maleic anhydride) (PEMA)-treated specimens was opposite to that on Epolene E-43 (a MAPP copolymer with a low molecular weight)-treated specimens. Based on these two models, the wetting slope and K value were used to interpret the kinetics of wetting. Therefore, these methods were helpful to characterize the dynamic wettability of wood surfaces modified with different coupling agents

Surface and Interfacial Characterization of Wood-PVC Composite: Imaging Morphology and Wetting Behavior1

An imaging technique was used to investigate wetting behavior of wood-PVC composites in this study. Two-dimensional and time-dependent profiles of water droplets on maleated wood surface and wood-PVC interface were observed. Experimental results indicated that coupling agents Epolene E-43 (a maleated polypropylene copolymer with low molecular weight)- and polyethylene-maleic anhydride (PEMA)- treated veneers had a hydrophilic surface, whereas the coupling agent Epolene G-3015 (a maleated polypropylene copolymer with high molecular weight)-treated veneer had a hydrophobic surface. For E-43- and PEMA-treated veneers, a water droplet had an elliptical shape after initial contact with the wood surface. However, a sessile droplet on G-3015-treated specimens was closer to a circular shape. During wetting, contact angle changes on E-43- and PEMA-treated specimens were larger than those on G-3015- treated specimens. Contact angles on maleated specimens with heat treatment and maleated interphases were almost independent of wetting time. Initial contact angle was influenced by coupling agent type, acid number, and retention and directions of wood grains. Initial contact angle decreased with increase of E-43 retention, but it was proportional to PEMA retention. However, it was independent of G-3015 retention. Wettability of fractured wood-PVC interface was similar to that of maleated wood surface with heat treatment. Thus, the interfacial characteristics of wood-PVC interface can be simulated with maleated wood surface with heat treatment

Selected Properties of Wood Strand and Oriented Strandboard From Small-Diameter Southern Pine Trees

Thermal and mechanical properties of southern pine and willow strands and properties of southern pine oriented strandboard (OSB) from small-diameter logs were investigated in this study. The effects of density and species group on tensile strength, dynamic moduli, and thermal stability of wood strands, and of strand quality (i.e., wood fines) on three-layer OSB properties were analyzed.Strand tensile strength and dynamic storage moduli (E') increased with the increase of strand density. A large variation in both tensile strength and E' values was observed for southern pine, while willow strands showed much smaller variability. The dynamic moduli (E") of strands decreased with increase of temperature in the range of 25° to 200°C. Small loss modulus (E") peaks were observed over the temperature range studied. The strands with higher densities had higher E". Thermogravimetric analysis results revealed that high-density strands were thermally more stable than low-density strands.Three-layer OSB made of small-diameter southern pine trees showed satisfactory strength and dimensional stability properties. As the fines loading levels increased, linear expansion (LE) along the parallel direction decreased, while the LE value along the perpendicular direction and thickness swelling increased. With increased fines levels, the internal bond strength showed an increasing trend up to the 20% fines level, and bending strength and modulus varied little in the parallel direction and slightly decreased in the perpendicular direction

Chemical Coupling in Wood Fiber and Polymer Composites: A Review of Coupling Agents and Treatments

Coupling agents in wood fiber and polymer composites (WFPC) play a very important role in improving the compatibility and adhesion between polar wood fibers and non-polar polymeric matrices. In this paper, we review coupling agents, pretrcatment, and mixing technology for wood fiber and polymer currently used in the manufacture of WFPC. So far, over forty coupling agents have been used in production and research. These agents are classified as organic, inorganic, and organic-inorganic groups, among which organic agents are better than inorganic agents because of stronger interfacial adhesion. The most popular coupling agents currently being used include isocyanates, anhydrides, silanes, and anhydride-modified copolymers. Coupling agents are usually coated on the surface of wood fiber, polymer or both by compounding, blending, soaking, spraying, or other methods. Three basic processes suitable for coupling treatment are discussed: directly coating during mixing and fully or partly pretreating before mixing. The pretreatment of wood fiber and polymer by coating or grafting is the preferred method to improve the mechanical properties of WFPC.