Szeged-type indices of subdivision vertex-edge join (SVE-join)

In this article, we compute the vertex Padmakar-Ivan (PIv ) index, vertex Szeged (Szv ) index, edge Padmakar-Ivan (PIe ) index, edge Szeged (Sze ) index, weighted vertex Padmakar-Ivan (wPIv ) index, and weighted vertex Szeged (wSzv ) index of a graph product called subdivision vertex-edge join of graphs.

On new algorithmic techniques for the weighted vertex coloring problem

The classical NP-hard weighted vertex coloring problem consists in minimizing the number of colors in colorings of vertices of a given graph so that, for each vertex, the number of its colors equals a given weight of the vertex and adjacent vertices receive distinct colors. The weighted chromatic number is the smallest number of colors in these colorings. There are several polynomial-time algorithmic techniques for designing efficient algorithms for the weighted vertex coloring problem. For example, standard techniques of this kind are the modular graph decomposition and the graph decomposition by separating cliques. This article proposes new polynomial-time methods for graph reduction in the form of removing redundant vertices and recomputing weights of the remaining vertices so that the weighted chromatic number changes in a controlled manner. We also present a method of reducing the weighted vertex coloring problem to its unweighted version and its application. This paper contributes to the algorithmic graph theory.