The Vertex Coloring Problem and its generalizations

The Vertex Coloring Problem and its generalizations

Polynomial Cases for the Vertex Coloring Problem

The computational complexity of the Vertex Coloring problem is known for all hereditary classes of graphs defined by forbidding two connected five-vertex induced subgraphs, except for seven cases. We prove the polynomial-time solvability of four of these problems: for (P5, dart)-free graphs, (P5, banner)-free graphs, (P5, bull)-free graphs, and (fork, bull)-free graphs.

The Frobenius Problem and Its Generalizations

The Generalized Assignment Problem and Its Generalizations

The generalized assignment problem is a classical combinatorial optimization problem that models a variety of real world applications including flexible manufacturing systems [6], facility location [11] and vehicle routing problems [2]. Given n jobs J = {1, 2, . . . , n} and m agents I = {1, 2, . . . ,m}, the goal is to determine a minimum cost assignment subject to assigning each job to exactl...

Edge-coloring Vertex-weightings of Graphs

Let $G=(V(G),E(G))$ be a simple, finite and undirected graph of order $n$. A $k$-vertex weightings of a graph $G$ is a mapping $w: V(G) to {1, ldots, k}$. A $k$-vertex weighting induces an edge labeling $f_w: E(G) to N$ such that $f_w(uv)=w(u)+w(v)$. Such a labeling is called an {it edge-coloring k-vertex weightings} if $f_{w}(e)not= f_{w}(echr(chr(chr('39')39chr('39'))39chr(chr('39')39chr('39'...