Abstract: The sort of question to be considered has the following form. Suppose we are given arecursive graph G, with vertex set V and edge set E. By “recursive” we mean (very roughly, but good enough for almost all purposes) that there is an effective algorithm which will allow us to compute in finitely many steps whether or not a givenx is in V, and also for given x,y to determine whether or not there is an edge in E joining them. For example, if G is finite, then it is recursive (but this case really is of no interest). It is usually quite safe to assume that V is just the set ω of natural numbers, or, perhaps, a recursive subset of ω. Next suppose that G is n-colorable, by which is meant that there is a function ϕ:V→ 0,1,2,…,n-l such that if x,y ∈V are adjacent, then ϕ (x) ≠ ϕ (y). Now the question: is G recursively n-colorable? I.e., is there a recursive n-coloring ϕ ofG? If not, is there some k for which there is a recursive k-coloring ofG? More generally, what conditions can be imposed on a recursive n-colorable graph which will guarantee that it is recursively k-colorable?

Abstract: In (strongly) perfect graphs, we define (strongly) canonical colorings; we show that for some classes of graphs, such colorings can be obtained by sequential coloring techniques. Chromatic properties ofP 4-free graphs based on such coloring techniques are mentioned and extensions to graphs containing no inducedP 5, $$\bar P_5 $$ orC 5 are presented. In particular we characterize the class of graphs in which any maximal (or minimal) nodex in the vicinal preorder has the following property: there is either noP 4 havingx as a midpoint or noP 4 havingx as an endpoint. For such graphs, according to a result of Chvatal, there is a simple sequential coloring algorithm.

Abstract: The Four Color Problem has probably been the most notorious mathematical problem of modern times. This problem asks whether four colors suffice to color every planar map so that adjacent countries, i.e., countries that share a border of positive length, receive different colors. This question's deceptive simplicity attracted many would-be solvers who oft times spent years on their search for a solution. Most returned empty-handed, or worse, with a false proof. Some were fortunate enough to have devised a new twist on the original problem that was sufficiently interesting to attract the attention of other aficionados. The Heawood Conjecture, one of the earliest of these offshoots, proved to be also one of the most fascinating and difficult. This other coloring conjecture guessed at the number of colors required by maps on other, more complicated, surfaces. Surprisingly enough, even though this later problem seems more difficult than its planar progenitor, Heawood's conjecture was actually verified a decade earlier. In my opinion this verification marks a milestone in the development of the modern combinatorial approach to geometry. It is my intention here to formulate this problem, recount its history, and discuss its relationship to the original Four Color Problem, as well as to other branches of mathematics.

Abstract: We define a natural probability distribution over the set of k-colorable graphs on n vertices and study the probable performance of several algorithms on graphs selected from this distribution. The main results are listed below. • We describe an algorithm to determine if a given n vertex graph is k-colorable, which runs in time O(n + m log k), where m is the number of edges. We show that this algorithm can successfully identify almost all random k-colorable graphs for constant or slowly growing values of k. • We show that an algorithm proposed by Brelas, and justified on... Read complete abstract on page 2.

Abstract: In this paper, two new concepts-Coloring-Contradiction (CC) and Non-Coloring-Contradiction Graph (NCCG) are introduced and some of their main properties are studied. The counting problem of NCCG is discussed, and a simple and efficient modification of the algorithm for coloring a graph is also presented.

Abstract: A generalized algorithm for graph coloring by implicit enumeration is formulated. A number of backtracking sequential methods are discussed in terms of the generalized algorithm. Some are revealed to be partially correct and inexact. A few corrections to the invalid algorithms, which cause these algorithms to guarantee optimal solutions, are proposed. Finally, some computational results and remarks on the practical relevance of improved implicit enumeration algorithms are given.

Abstract: A graph is cubical if it is a subgraph of a hypercube; the dimension of the smallest such hypercube is the dimension of the graph. We show several results concerning this class of graphs. We use a characterization of cubical graphs in terms of edge coloring to show that the dimension of biconnected cubical graphs is at most half the number of nodes. We also show that telling whether a graph is cubical is NP-complete. Finally, we propose a heuristic for minimizing the dimension of trees, which yields an embedding of the tree in a hypercube of dimension at most the square of the true dimension of the tree.