Graph theory I
An algorithm that has successfully and efficiently 4-colored all planar graphs attempted is described, generally utilizing from i0 to 20 seconds of 360/50 CPU time for maximal planar vertices, and efficiency analysis of all known algorithms for finding all circuits is made.
Abstract
Univ.-We describe an algorithm that has successfully and efficiently 4-colored all planar graphs attempted, generally utilizing from i0 to 20 seconds of 360/50 CPU time for maximal planar graphs of 200-500 vertices. A modified recursive-smallest-degree-last I ordering of the vertices is first determined, and then a sequential coloring with bichromatic interchange I is utilized. Our new algorithm utilizes additional steps suggested by some recent theory attendant to insertion of a vertex of degree 5. We also describe an algorithm for generating pseudo-random maximal planar graphs of minimum degree 5 with optional controls to avoid generating 3 and 4 connected graphs. Statistics of the application of the coloring algorithm to such randomly generated planar graphs and other test graphs will be summarized. efficiency analysis of all known algorithms for finding all circuits is made, and bounds on the computation time are derived. I Best algorithms are exhibited. The vector-space method of circuit generation is discussed at length. Proven is the fact that the following are the only ones which have 2~-i circuits (~ is the nullity of the graph), or, equivalently, have no two edge-disjoint circuits. A class of K3 ~4-x K4 ~3,3 graphs, circuit-graphs, derivable out of a graph are defined. If an efficient algorithm for the enumeration of all connected, induced subgraphs is known, these circuit-graphs can advantageously be used in the generation of circuits. LA/a~ Bounds on the Weighted Path Length of Binary Trees, JI~AN PRADEI~ U. of Illinois-The weighted path length of binary trees is a quantity of interest in ma/ly areas, such as information storage and retrieval, sorting and searching, and coding theory~ it measures such important quantities as the average search-or processing-time, or the average length of code words. Various bounds of this important quantity are derived. They are given by relatively simple expression and yet are reasonably tight over a wide range of shapes of trees and distribution of weights. W~ ~ On the Chromatic Number of a Graoh. CHUNG C. WANG, State University of New York at Buffalo-Recently Christofides' describes on a~~m for coloring the vertices of a graoh using the minim~n number of colors so that no two adjacent vertices are colored the same. His algorithm implies the construction of a particular subgraph tree for a given graph and finding, by a breadth-first search rule, a shortest path between the root and terminal nodes of this subgraph tree. The breadth of the …