### On-line Ranking of Split Graphs

"... A 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 algorith ..."

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A 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−ε)-competitive semi on-line algorithm of this type. Finally, a 2-competitive semi on-line algorithm is given.

### and

"... A parity path in a vertex colouring of a graph is a path along which each colour is used an even number of times. Let χp(G) be the least number of colours in a proper vertex colouring of G having no parity path. It is proved that for any graph G we have the following tight bounds χ(G) ≤ χp(G) ≤ |V ..."

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A parity path in a vertex colouring of a graph is a path along which each colour is used an even number of times. Let χp(G) be the least number of colours in a proper vertex colouring of G having no parity path. It is proved that for any graph G we have the following tight bounds χ(G) ≤ χp(G) ≤ |V (G) | − α(G) + 1, where χ(G) and α(G) are the chromatic number and the independence number of G, respectively. The bounds are improved for trees. Namely, if T is a tree with diameter diam(T) and radius rad(T), then log2

### CONTENTS Preliminary Program................................................... 7 Abstracts

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### CONTENTS Preliminary Program................................................... 7 Abstracts

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### ON EXTREMAL SIZES OF LOCALLY k-TREE GRAPHS

, 2008

"... On extremal sizes of locally k-tree graphs ..."

### Abstracts of Invited Lectures On refinable functions, subdivision and wavelets

"... An introduction is given to the mathematical analysis of the related concepts of refinable functions, subdivision and wavelets. Iterative methods based on subdivision are widely applied in geometric modelling, whereas wavelet decomposition techniques represent a viable alternative to Fourier methods ..."

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An introduction is given to the mathematical analysis of the related concepts of refinable functions, subdivision and wavelets. Iterative methods based on subdivision are widely applied in geometric modelling, whereas wavelet decomposition techniques represent a viable alternative to Fourier methods in signal analysis. Of fundamental importance in both subdivision and wavelet analysis is the issue of the existence of a solution, called a refinable function, of a certain refinement equation. We demonstrate the link between subdivision convergence and refinable function existence, and we show how a given refinable function can be used as a building block for the multiresolutional method of wavelet construction. Also, we present an overview of refinable function results with respect to existence and uniqueness, regularity (or smoothness), as well as a variety of other properties. An algebraic view of weaker forms of realcompactness

### British Library Cataloguing in Publication Data

"... A catalogue record for this book is available from the British Library ..."

### Industrial bioprocesses and biocatalysis Lectures L1.1

"... Starch is an interesting, natural, abundant and renewable raw material. For many applications, its properties can be improved by chemical modification, and acylation has been commonly used. Traditionally this makes use of powerful acylating agents (anhydrides and acid chlorides) and/or harsh conditi ..."

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Starch is an interesting, natural, abundant and renewable raw material. For many applications, its properties can be improved by chemical modification, and acylation has been commonly used. Traditionally this makes use of powerful acylating agents (anhydrides and acid chlorides) and/or harsh conditions. Enzymatic acylation would be attractive because of the use of mild conditions, specific reaction, and the avoidance of by-products, particularly if fatty acids can be used as acyl donors. It is difficult to find a reaction system in which the three species can be brought into mutual contact: the starch; an enzyme; and water insoluble fatty acids or derivatives. However, there are some literature reports of successful acylation in both organic and aqueous gel media. This talk will present our recent studies of acylation with medium chain fatty acids, mainly with the liquid