a modular $k$-coloring, $kge 2,$ of a graph $g$ without isolated vertices is a coloring of the vertices of $g$ with the elements in $mathbb{z}_k$ having the property that for every two adjacent vertices of $g,$ the sums of the colors of the neighbors are different in $mathbb{z}_k.$ the minimum $k$ for which $g$ has a modular $k-$coloring is the modular chromatic number of $g.$ except for some s...

let $g$ be a connected graph of order $3$ or more and $c:e(g)rightarrowmathbb{z}_k$‎ ‎($kge 2$) a $k$-edge coloring of $g$ where adjacent edges may be colored the same‎. ‎the color sum $s(v)$ of a vertex $v$ of $g$ is the sum in $mathbb{z}_k$ of the colors of the edges incident with $v.$ the $k$-edge coloring $c$ is a modular $k$-edge coloring of $g$ if $s(u)ne s(v)$ in $mathbb{z}_k$ for all pa...

Two vertices u and v in a nontrivial connected graph G are twins if u and v have the same neighbors in V (G) − {u, v}. If u and v are adjacent, they are referred to as true twins; while if u and v are nonadjacent, they are false twins. For a positive integer k, let c : V (G) → Zk be a vertex coloring where adjacent vertices may be assigned the same color. The coloring c induces another vertex c...

A "first" implementation of the Modular Smalltalk object-oriented programming language is presented. The implementation includes an object-oriented parser, object-oriented representation for code fragments and an object-oriented C-code generator, all implemented in Smalltalk-80. This implementation validates two of the five design principles of the Modular Smalltalk language and provides a vehi...

Partitive set families are families of sets that can be quite large, but have a compact, recursive representation in the form of a tree. This tree is a common generalization of PQ trees, the modular decomposition of graphs, certain decompositions of boolean functions, and decompositions that arise on a variety of other combinatorial structures. We describe natural operators on partitive set fam...

Let G be a graph. A proper vertex coloring of G is said to be a dynamic coloring if for every v ∈ V (G) of degree at least 2, the neighbors of v receive at least two different colors. The smallest integer k such that G has a dynamic k-coloring is called the dynamic chromatic number of G and is denoted by χ2(G). It was conjectured that if G is an r-regular graph, then χ2(G)−χ(G) ≤ 2. In this tal...

Cographs are exactly the hereditarily well-colored graphs, i.e., graphs for which a greedy vertex coloring of every induced subgraph uses only minimally necessary number colors χ(G). Greedy colorings shown to be subclass hierarchical that naturally appear in phylogenetic combinatorics. The cographs form special class minimal colorings. fact cotrees modular decomposition trees suggests natural g...

the robust coloring problem (rcp) is a generalization of the well-known graph coloring problem where we seek for a solution that remains valid when extra edges are added. the rcp is used in scheduling of events with possible last-minute changes and study frequency assignments of the electromagnetic spectrum. this problem has been proved as np-hard and in instances larger than 30 vertices, meta-...

It is known that a number of natural graph problems which are FPT parameterized by treewidth become W-hard when parameterized by clique-width. It is therefore desirable to find a different structural graph parameter which is as general as possible, covers dense graphs but does not incur such a heavy algorithmic penalty. The main contribution of this paper is to consider a parameter called modul...