Here presented is a unified expression of Stirling numbers and their generalizations by using generalized factorial functions and generalized divided difference. Previous well-known extensions of Stirling numbers due to Riordan, Carlitz, Howard, Charalambides-Koutras, Gould-Hopper, Hsu-Shiue, Tsylova, Todorov, and Ahuja-Enneking are included as particular
cases of our generalization. Four algorithms for calculating the Stirling numbers and their generalizations based on our unified form are also given, which include two comprehensive algorithms using the characterization of Riordan arrays.

We give necessary and sufficient conditions to decompose \( \lambda \) copies, where necessarily \( \lambda \geq 2 \), of the complete graph \( K_v \), into so-called “2-petal”, “stem-infinity”, “barbell”, and “box-edge” graphs, all with four vertices and five edges.

The total chromatic number conjecture, which has appeared in a few hundred articles and in numerous books thus far, is now one of the classic mathematical unsolved problems. It appears that many authors coincidentally have attributed it to Professor M. Behzad and/or to Professor V.G. Vizing. Eventually, after four decades, Professor A. Soifer investigated the origin of this conjecture; published his findings in *The Mathematical Coloring Book* (2009); and stated that, “In my opinion this unquestionably merits the joint credit to Vizing and Behzad.” After checking all the arguments presented and the blames cited, I decided to investigate the controversy stated in this book on my own. My findings, which are presented in this report, specifically signify the following two points:

1. M. Behzad is the sole author of the Total Chromatic Number Conjecture.
2. The wrong referrals provided by numerous authors over the last forty-four years, to indicate Vizing’s authorship, must be brought to the attention of the authors and researchers, by appropriate means, as soon as possible.

Let \(\mathcal{G}_n\) be the set of all simple loopless undirected graphs on \(n\) vertices. Let \(T\) be a linear mapping, \(T : \mathcal{G}_n \rightarrow \mathcal{G}_n\) for which the independence number of \(T(G)\) is the same as the independence number for \(G\) for any \(G \in \mathcal{G}_n\). We show that \(T\) is necessarily a vertex permutation. Similar results are obtained for mappings preserving the matching number of bipartite graphs, the vertex cover number of undirected graphs, and the edge independence number of undirected graphs.