On-line -coloring of graphs

For a given induced hereditary property , a -coloring of a graph G is an assignment of one color to each vertex such that the subgraphs induced by each of the color classes have property . We consider the effectiveness of on-line -coloring algorithms and give the generalizations and extensions of selected results known for on-line proper coloring algorithms. We prove a linear lower bound for the performance guarantee function of any stingy on-line -coloring algorithm. In the class of generalized trees, we characterize graphs critical for the greedy -coloring. A class of graphs for which a greedy algorithm always generates optimal -colorings for the property = K₃-free is given

On extremal sizes of locally kk-tree graphs

summary:A graph $G$ is a {\it locally $k$-tree graph} if for any vertex $v$ the subgraph induced by the neighbours of $v$ is a $k$-tree, $k\ge 0$, where $0$-tree is an edgeless graph, $1$-tree is a tree. We characterize the minimum-size locally $k$-trees with $n$ vertices. The minimum-size connected locally $k$-trees are simply $(k+1)$-trees. For $k\ge 1$, we construct locally $k$-trees which are maximal with respect to the spanning subgraph relation. Consequently, the number of edges in an $n$-vertex locally $k$-tree graph is between $\Omega (n)$ and $O(n^2)$, where both bounds are asymptotically tight. In contrast, the number of edges in an $n$-vertex $k$-tree is always linear in $n$

Partitions of some planar graphs into two linear forests

A linear forest is a forest in which every component is a path. It is known that the set of vertices V(G) of any outerplanar graph G can be partitioned into two disjoint subsets V₁,V₂ such that induced subgraphs ⟨V₁⟩ and ⟨V₂⟩ are linear forests (we say G has an (LF, LF)-partition). In this paper, we present an extension of the above result to the class of planar graphs with a given number of internal vertices (i.e., vertices that do not belong to the external face at a certain fixed embedding of the graph G in the plane). We prove that there exists an (LF, LF)-partition for any plane graph G when certain conditions on the degree of the internal vertices and their neighbourhoods are satisfied

-bipartitions of minor hereditary properties

We prove that for any two minor hereditary properties ₁ and ₂, such that ₂ covers ₁, and for any graph G ∈ ₂ there is a ₁-bipartition of G. Some remarks on minimal reducible bounds are also included

On-line ranking of split graphs

Graphs and AlgorithmsA vertex ranking of a graph G is an assignment of positive integers (colors) to the vertices of G such that each path connecting two vertices of the same color contains a vertex of a higher color. Our main goal is to find a vertex ranking using as few colors as possible. Considering on-line algorithms for vertex ranking of split graphs, we prove that the worst case ratio of the number of colors used by any on-line ranking algorithm and the number of colors used in an optimal off-line solution may be arbitrarily large. This negative result motivates us to investigate semi on-line algorithms, where a split graph is presented on-line but its clique number is given in advance. We prove that there does not exist a (2-ɛ)-competitive semi on-line algorithm of this type. Finally, a 2-competitive semi on-line algorithm is given

Graph Classes Generated by Mycielskians

In this paper we use the classical notion of weak Mycielskian M′(G) of a graph G and the following sequence: M′0(G) = G, M′1(G) = M′(G), and M′n(G) = M′(M′n−1(G)), to show that if G is a complete graph of order p, then the above sequence is a generator of the class of p-colorable graphs. Similarly, using Mycielskian M(G) we show that analogously defined sequence is a generator of the class consisting of graphs for which the chromatic number of the subgraph induced by all vertices that belong to at least one triangle is at most p. We also address the problem of characterizing the latter class in terms of forbidden graphs

Expanding Access to Optically Active Non-Steroidal Anti-Inflammatory Drugs via Lipase-Catalyzed KR of Racemic Acids Using Trialkyl Orthoesters as Irreversible Alkoxy Group Donors

Studies into the enzymatic kinetic resolution (EKR) of 2-arylpropanoic acids (‘profens’), as the active pharmaceutical ingredients (APIs) of blockbuster non-steroidal anti-inflammatory drugs (NSAIDs), by using various trialkyl orthoesters as irreversible alkoxy group donors in organic media, were performed. The enzymatic reactions of target substrates were optimized using several different immobilized preparations of lipase type B from the yeast Candida antarctica (CAL-B). The influence of crucial parameters, including the type of enzyme and alkoxy agent, as well as the nature of the organic co-solvent and time of the process on the conversion and enantioselectivity of the enzymatic kinetic resolution, is described. The optimal EKR procedure for the racemic profens consisted of a Novozym 435-STREM lipase preparation suspended in a mixture of 3 equiv of trimethyl or triethyl orthoacetate as alkoxy donor and toluene or n-hexane as co-solvent, depending on the employed racemic NSAIDs. The reported biocatalytic system provided optically active products with moderate-to-good enantioselectivity upon esterification lasting for 7–48 h, with most promising results in terms of enantiomeric purity of the pharmacologically active enantiomers of title APIs obtained on the analytical scale for: (S)-flurbiprofen (97% ee), (S)-ibuprofen (91% ee), (S)-ketoprofen (69% ee), and (S)-naproxen (63% ee), respectively. In turn, the employment of optimal conditions on a preparative-scale enabled us to obtain the (S)-enantiomers of: flurbiprofen in 28% yield and 97% ee, ibuprofen in 45% yield and 56% ee, (S)-ketoprofen in 23% yield and 69% ee, and naproxen in 42% yield and 57% ee, respectively. The devised method turned out to be inefficient toward racemic etodolac regardless of the lipase and alkoxy group donor used, proving that it is unsuitable for carboxylic acids possessing tertiary chiral centers