A {\it star-factor} of a graph $G$ is a spanning subgraph of $G$ such that each of its component is a star. Clearly, every graph without isolated vertices has a star factor. A graph $G$ is called {\it star-uniform} if all star-factors of $G$ have the same number of components. To characterize star-uniform graphs was an open problem posed by Hartnell and Rall, which is motivated by the minimum cost spanning tree and the optimal assignment problems. We use the concepts of factor-criticality and domination number to characterize all star-uniform graphs with the minimum degree at least two. Our proof is heavily relied on Gallai-Edmonds Matching Structure Theorem.
A {\it star-factor} of a graph $G$ is a spanning subgraph of $G$ such that each of its component is a star. Clearly, every graph without isolated vertices has a star factor. A graph $G$ is called {\it star-uniform} if all star-factors of $G$ have the same number of components. To characterize star-uniform graphs was an open problem posed by Hartnell and Rall, which is motivated by the minimum cost spanning tree and the optimal assignment problems. We use the concepts of factor-criticality and domination number to characterize all star-uniform graphs with the minimum degree at least two. Our proof is heavily relied on Gallai-Edmonds Matching Structure Theorem.
Proposed as a general framework, Liu and Yu(Discrete Math. 231 (2001) 311-320) introduced $(n,k,d)$-graphs to unify the concepts of deficiency of matchings, $n$-factor-criticality and $k$-extendability. Let $G$ be a graph and let $n,k$ and $d$ be non-negative integers such that $n+2k+d\leq |V(G)|-2$ and $|V(G)|-n-d$ is even. If when deleting any $n$ vertices from $G$, the remaining subgraph $H$ of $G$ contains a $k$-matching and each such $k$- matching can be extended to a defect-$d$ matching in $H$, then $G$ is called an $(n,k,d)$-graph. In \cite{Liu}, the recursive relations for distinct parameters $n, k$ and $d$ were presented and the impact of adding or deleting an edge also was discussed for the case $d = 0$. In this paper, we continue the study begun in \cite{Liu} and obtain new recursive results for $(n,k,d)$-graphs in the general case $d \geq0$.
Wood surface roughness, surface free energy (SFE), wettability, and bonding quality for water-based acrylic coatings were investigated. The samples tested in this study included Pinus radiata, Pinus sylvestris, Larch, Hemp oak, Catalpa tree, and Camphor. Sandpaper with grits of 180, 240, 320, 400, and 500 was utilized to sand wood surfaces. The van OSS-Chaudhury-Good equation (vOCG) was used to calculate the SFE values. The modified model (M-D) was used to calculate the wettability based on the contact angle change rate (K value). The higher the K value, the faster the contact angle approaches equilibrium. A cross-cut test was used to evaluate the coating’s bonding quality. The anatomical structure of wood has an impact on the roughness of hardwood. The equilibrium contact angle is influenced by the wood species and sandpaper grit size. Sanding can make the surface of wood more wettable. Radiata pine that had been sanded to 180 grit had the highest SFE value. After finishing with waterborne acrylic, hardwood had a slightly better coating adhesion than softwood. Hemp oak wood had the lowest coating adhesion (0.6) and the highest K value (0.82). The best bonding quality (0.4) was supplied by the camphor wood with the lowest K value (0.13). Wettability in terms of K values was a good indication of determining the bonding quality of the water-based acrylic coatings.
Let s,m,n be positive integers and F be a graph. A graph G is called F-saturated if F is not a subgraph of G but G + e contain a copy of F for every edge e ∈ E(G), where G is the complement graph of G. Let sat(n,F) be the minimum number of edges over all F-saturated graphs with order n and Sat(n,F) denotes the family of F-saturated graphs with sat(n,F) edges and n vertices. Let mK2 denote a matching of size m and let denote a fractional matching of size
'/%3e%3cuse xlink:href='%23a' transform='rotate(23.036 .093 25.536)'/%3e%3cuse xlink:href='%23a' transform='rotate(45.87 1.273 16.18)'/%3e%3cuse xlink:href='%23a' transform='rotate(69.945 .996 12.078)'/%3e%3cuse xlink:href='%23a' transform='rotate(20.66 -19.689 31.932)'/%3e%3c/svg%3e)