Let G be a series-parallel graph. In this paper, we present a linear algorithm of constructing an oriented binary decomposition tree of G. We use it to find 33 unavoidable subgraphs of G. Based on these 33 avoidable subgraphs, we can determine the edge-face chromatic number, denoted by χef (G), of G where G is 2-connected and Δ(G) = 5. This completes the literature of determining χef (G) for 2-...

We propose two new self-stabilizing distributed algorithms for proper +1 ( is the maximum degree of a node in the graph) coloring of arbitrary system graphs. Both algorithms are capable of working with multiple types of demons (schedulers) as is the most recent algorithm in [1]. The first algorithm converges in O(m) moves while the second converges in at most n moves (n is the number of nodes a...

A feedback neural network for solving graph coloring problem is presented. The circuit has an associated transcendental energy function that ensures fast convergence to the exact solution. Hardware and PSPICE simulation results on random and benchmark problems have been presented. Test results are compared with existing techniques for graph coloring to show that the proposed neural network mode...

The vertex coloring problem asks for the minimum number of colors that can be assigned to the vertices of a given graph such that for all vertices v the color of v is different from the color of any of its neighbors. The problem is NP-hard. Here, we introduce new integer linear programming formulations based on partial orderings. They have the advantage that they are as simple to work with as t...

A b s t r a c t . Many combinatorial problems can be efficiently solved for partial k-trees. The edge-coloring problem is one of a few combinatorial problems for which no linear-time algorithm has been obtained for partial k-trees. The best known algorithm solves the problem for partial k-trees G in time O(nA 2~r where n is the number of vertices and A is the maximum degree of G. This paper giv...

The distributed (∆ + 1)-coloring problem is one of most fundamental and well-studied problems in Distributed Algorithms. Starting with the work of Cole and Vishkin in 86, there was a long line of gradually improving algorithms published. The current state-of-the-art running time is O(∆ log ∆+log∗ n), due to Kuhn and Wattenhofer, PODC’06. Linial (FOCS’87) has proved a lower bound of 1 2 log∗ n f...

This paper examines a parameterized problem that we refer to as n− k Graph Coloring, i.e., the problem of determining whether a graph G with n vertices can be colored using n−k colors. As the main result of this paper, we show that there exists a O(kn+k+2) = O(n) algorithm for n− k Graph Coloring for each fixed k. The core technique behind this new parameterized algorithm is kernalization via m...

We study the complexity of several coloring problems on graphs, parameterized by the treewidth t of the graph: (1) The list chromatic number χl(G) of a graph G is defined to be the smallest positive integer r, such that for every assignment to the vertices v of G, of a list Lv of colors, where each list has length at least r, there is a choice of one color from each vertex list Lv yielding a pr...