Showing papers in "Journal of Graph Theory in 1977"

The graph isomorphism disease

Abstract: The graph isomorphism problem—to devise a good algorithm for determining if two graphs are isomorphic—is of considerable practical importance, and is also of theoretical interest due to its relationship to the concept of NP-completeness. No efficient (i.e., polynomial-bound) algorithm for graph isomorphism is known, and it has been conjectured that no such algorithm can exist. Many papers on the subject have appeared, but progress has been slight; in fact, the intractable nature of the problem and the way that many graph theorists have been led to devote much time to it, recall those aspects of the four-color conjecture which prompted Harary to rechristen it the “four-color disease.” This paper surveys the present state of the art of isomorphism testing, discusses its relationship to NP-completeness, and indicates some of the difficulties inherent in this particularly elusive and challenging problem. A comprehensive bibliography of papers relating to the graph isomorphism problem is given.

Graph reconstruction—a survey

Abstract: The Reconstruction Conjecture asserts that every finite simple undirected graph on three or more vertices is determined, up to isomorphism, by its collection of vertex-deleted subgraphs. This article reviews the progress made on the conjecture since it was first formulated in 1941 and discusses a number of related questions.

Extremal problems in graph theory

Abstract: The aim of this note is to give an account of some recent results and state a number of conjectures concerning extremal properties of graphs

Tree‐complete graph ramsey numbers

Abstract: The ramsey number of any tree of order m and the complete graph of order n is 1 + (m − 1)(n − 1).

The number of connected sparsely edged graphs

Abstract: An (n, q) graph has n labeled points, q edges, and no loops or multiple edges. The number of connected (n, q) graphs is f(n, q). Cayley proved that f(n, n-1) = nn−2 and Renyi found a formula for f(n, n). Here I develop two methods to calculate the exponential generating function of f(n, n + k) for particular k and so to find a formula for f(n, n + k) for general n. The first method is a recurrent one with respect to k and is well adapted for machine computation, but does not itself provide a proof that it can be continued indefinitely. The second (reduction) method is much less efficient and is indeed impracticable for k greater than 2 or 3, but it supplies the missing proof that the generating function is of a particular form and so that the first method can be continued for all k, subject only to the capacity of the machine.

The falsity of the reconstruction conjecture for tournaments

Abstract: The conjecture that for all sufficiently large p any tournament of order p is uniquely reconstructable from its point-deleted subtournaments is shown to be false. Counterexamples are presented for all orders of the form 2n + 1 and 2n + 2. The largest previously known counterexamples were of order 8.

A note of welcome

Abstract: A note of welcome to the new Journal of Graph Theory might contain all sorts of good wishes and superficial praises of the beauty and usefulness of graph theory in general terms. My views on the latter, supported by facts, were given in [2]. As to the former, I can illustrate it better by giving some indications of the enchantment and help it gave me in the most difficult times of my life during the war. It sounds a bit incredible but it is true. The story goes back to 1940 when I received a letter from Shanghai from my friend George Szekeres in which he described an unsuccessful attempt to prove a famous Burnside conjecture (which was disproved later). The failure of his attempt could be effected by a special case of Ramsey’s theorem (but Ramsey’s paper, beyond mere existence, was unknown at that time in Hungary). At that time my main financial income came from private tutoring, and I had to teach the pupils at their homes. After receiving the letter, and while traveling between two consecutive pupils, I was pondering on its content. The chain of thought soon led me to finite forms and then to the following extremal problem: What is the maximum number of edges in a graph with n vertices not containing a complete subgraph with k vertices? Though I found the problem definitely interesting, I postponed it, having been interested at that time mainly in problems in analytical number theory. In September 1940 I was called in for the first time to labor-camp service. We were taken to Transylvania to work at railway building. Our main work was carrying railway ties. It was not very difficult work but a spectator could of course easily recognize that most of us-I was no exception-did it rather awkwardly. One of my more expert comrades said this at one occasion quite explicitly, even mentioning my name. An officer was standing nearby, watching our work. When hearing my name, he asked the comrade whether or not I was a mathematician. It turned out that the officer-Joseph Winkler by namewas an engineer. In his youth he had placed at a mathematical competition; in civilian life he was a proofreader at the printing shop where the periodical of the Third Class of the Academy (Mathematical and Natural Sciences) was printed and

On the distance matrix of a directed graph

Abstract: In this note, we show how the determinant of the distance matrix D(G) of a weighted, directed graph G can be explicitly expressed in terms of the corresponding determinants for the (strong) blocks Gi of G. In particular, when cof D(G), the sum of the cofactors of D(G), does not vanish, we have the very attractive formula .

The spanning subgraphs of eulerian graphs

Abstract: It is shown that a connected graph G spans an eulerian graph if and only if G is not spanned by an odd complete bigraph K(2m + 1, 2n + 1). A disconnected graph spans an eulerian graph if and only if it is not the union of the trivial graph with a complete graph of odd order. Exact formulas are obtained for the number of lines which must be added to such graphs in order to get eulerian graphs.

The biparticity of a graph

Abstract: The biparticity β(G) of a graph G is the minimum number of bipartite graphs required to cover G. It is proved that for any graph G, β(G) = {log2χ(G)}. In view of the recent announcement of the Four Color Theorem, it follows that the biparticity of every planar graph is 2.

Some small ramsey numbers

Abstract: In previous work, the Ramsey numbers have been evaluated for all pairs of graphs with at most four points. In the present note, Ramsey numbers are tabulated for pairs F1, F2 of graphs where F1 has at most four points and F2 has exactly five points. Exact results are listed for almost all of these pairs.

Every (p, p-2) graph is contained in its complement†

Abstract: The following is proved: If G is graph of order p (≥2) and size p-2, then there exists an isomorphic embedding of G into its complement.

Blocks and the nonorientable genus of graphs

Abstract: Examples are given to show that the nonorientable genus of a graph is not additive over its blocks. A nonorientable analog for the Battle, Harary, Kodama, and Youngs Theorem is proved; this completely determines the nonorientable genus of a graph in terms of its blocks. It is also shown that one of the above counterexamples has the minimum possible order.

Total matchings and total coverings of graphs

Abstract: In graph theory, the related problems of deciding when a set of vertices or a set of edges constitutes a maximum matching or a minimum covering have been extensively studied. In this paper we generalize these ideas by defining total matchings and total coverings, and show that these sets, whose elements in general consist of both vertices and edges, provide a way to unify these concepts. Parameters denoting the maximum and the minimum cardinality of these sets are introduced and upper and lower bounds depending only on the order of the graph are obtained for the number of elements in arbitrary total matchings and total coverings. Precise values of all the parameters are found for several general classes of graphs, and these are used to establish the sharpness of most of the bounds. In addition, variations of some well known equalities due to Gallai relating covering and matching numbers are obtained.

A class of Hamiltonian polytopes

Abstract: It is shown that any simple 3-polytope, all of whose faces are triangles or hexagons, admits a hamiltonian circuit.

Some applications of graph contractions

Abstract: Results in diverse areas, such as the Nielsen-Schreier theorem on subgroups of free groups and a proof of A. T. White's conjecture on the genus of subgroups are shown to be immediate consequences of a lemma which has already proved useful in investigating topological properties and automorphism group of graphs.

A canonical representation of trivalent hamiltonian graphs

Abstract: A canonical representation of trivalent hamiltonian graphs in the form of “span lists” had been proposed by J. Lederberg. It is here presented in a modified form due to H. S. M. Coxeter and the author, and therefore called “LCF notation.” This notation has the advantage of being more concise than Lederberg's original span lists whenever the graph has a hamiltonian circuit with rotational symmetry. It is also useful as a method for a systematic classification of trivalent hamiltonian graphs and allows one to define for such graphs two interesting properties, called, respectively, “antipalindromic” and “quasiantipalindromic.”.

The combinatorial map color theorem

Abstract: This paper is written in the spirit of the author's book: Map Color Theorem (1974). We try to develop the Map Color Theorem in a combinatorial way, circumventing the unwieldy embedding theory. Similar (but not identical) generalizations have recently and independently been developed by Alpert (in press) and by Stahl (in press). The first nine theorems are one-dimensional versions of known facts from the theory of two-dimensional compact manifolds. Theorems 10 to 13 are to my knowledge completely new results.

Graphs with exactly one hamiltonian circuit

Abstract: Let h(n) be the largest integer such that there exists a graph with n vertices having exactly one Hamiltonian circuit and exactly h(n) edges. We prove that h(n) = [n2/4]+1 (n ≧ 4) and discuss some related problems.

Computer Reconstruction of Small Graphs

Abstract: The Reconstruction Conjecture is established for graphs with nine ver-tices.

On induced subgraphs of a block

Abstract: If G is a block, then a vertex u of G is called critical if G - u is not a block. In this article, relationships between the localization of critical vertices and the localization of vertices of relatively small degrees (especially, of degree two) are studied. A block is called semicritical if a) each edge is incident with at least one critical vertex and b) each vertex of degree two is critical. Let G be a semicritical block with at least six vertices. It is proved that A) there exist distinct vertices u2, v1, u2, and v2 of degree two in G such that u1v1 and u2v2 are edges of G, and u1v2, and u2v2 are edges of the complement of G, and B) the complement of G is a block with no critical vertex of degree two.

Packing and covering constants for certain families of trees. I

Abstract: If ℱ denotes a family of rooted trees, let pk(n) and ck(n) denote the average value of the k-packing and k-covering numbers of trees in ℱ that have n nodes. We assume, among other things, that the generating function y of trees in ℱ satisfies a relation of the type y = xϕ(y) for some power series ϕ. We show that the limits of pk(n)/n and ck(n)/n as n ∞ exist and we describe how to evaluate these limits.

Enumeration of graphs with signed points and lines

Abstract: Our object is to enumerate graphs in which the points or lines or both are assigned positive or negative signs. We also treat several associated problems for which these configurations are self-dual with respect to sign change. We find that the solutions to all of these counting problems can be expressed as special cases of one general formula involving the concatenation of the cycle index of the symmetric group with that of its pair group. This counting technique is based on Polya's Enumeration Theorem and the Power Group Enumeration Theorem. Using a suitable computer program, we list the number of graphs of each type considered up to twelve points. Sharp asymptotic estimates are also obtained.

An attempt to understand the four color problem

Abstract: Is the recently obtained, computer-aided proof of the Four Color Theorem an isolated phenomenon or is its combinatorial complexity typical for a significantly large class of mathematical problems? While it is too early to give a definite answer to this question, an informal discussion is undertaken in this article.

Some recent results in hamiltonian graphs

Abstract: A variety of recent developments in hamiltonian theory are reviewed. In particular, several sufficient conditions for a graph to be hamiltonian, certain hamiltonian properties of line graphs, and various hamiltonian properties of powers of graphs are discussed. Furthermore, the concept of an n-distant hamiltonian graph is introduced and several theorems involving this special class of hamiltonian graphs are presented.

A digest of the four color theorem

Abstract: A major event in 1976 was the announcement that the Four Color Conjecture (4CC) had at long last become the Four Color Theorem (4CT) The proof by W Haken, K Appel, and J Koch is published in the Illinois Journal of Mathematics, and their two-part article outlines the nature and reliability of the solution The first section is a readable and informative historical survey The reminder will appeal chiefly to specialists in graph theory Although the logic of attack is relatively simple, the need to examine an immense number of individual cases is frustrating Hopefully this first breakthrough will pave the way for a short elegant proof For the second section, 1200 hours of computer time was required to verify the 4-color reducibility of nearly 1900 configurations At this time there is no good way to condense the proof In this digest we offer an exposition of the main ideas The first and the last parts are intended for a general audience, but the intermediate sections assume more knowledge of graph theory proper The usual statement of the 4CC goes as follows: “All maps on the sphere or plane can be colored with four colors so that neighboring regions are never colored alike” The form in which the 4CT was proved is “There exists an unavoidable set of reducible configurations, relative to triangulations of the plane” The initial part of our task is to explain how the 4CC comes to be expressed in such jargon The next step is to show how one finds simple finite sets of unavoidable configurations Then comes the question of how to prove reducibility, followed by a consideration of the known obstacles to reduction Our concluding remarks and criticisms include a consideration of prospects for the future

Definitions of criticality with respect to edge-coloring

Abstract: Here we examine six definitions of criticality concerning the chromatic index (edge chromatic number) of a simple graph. Five of these turn out to be almost always almost equivalent. Some problems arise and some conjectures are posed.

Counting cubic graphs

Abstract: Unlabeled cubic graphs are enumerated. Labeled cubic graphs have already been counted by Read, in two different ways, one very elegant theoretically [3] and the other quite efficient computationally [4]. Our solution for the unlabeled problem, while not as elegant or efficient, will enable numerical results to be obtained by machine computation. The essential structural fact which we use for counting cubic graphs is that a graph is regular of degree 3 only if it has p = 2n points, q = 3n lines, and no point of degree less than 3. We first enumerate all unlabeled homeomorphically irreducible (p, q)-graphs without isolates or endpoints. Then the cubic graphs are just those with parameters ( 2 4 3n) . For labeled graphs the inclusion-exclusion principle has been used to count those without endpoints [4] and those without points of degree 2 by Jackson and Reilly [2]. For unlabeled graphs it is trivial to count those without isolates. Yet the only known method of counting graphs without endpoints requires the cycle index sum methods of [5]. The sum of the cyclic indices of the automorphism groups of a set of graphs contains the information needed to count them, as noted in Harary and Palmer [l, p. 1841, and a great deal more besides. In order to present the formula for counting graphs without endpoints let Z ( W ) be the sum of the cycle indices of the point automorphism groups of the nontrivial connected graphs without endpoints, and let Z ( C ) , Z ( V , and Z(T') be the known cycle index sums for all connected graphs, trees, and rooted trees, respectively. The composition of two cycle index sums is a natural extension of that of the composition of two cycle indices as defined in [l, p. 981. Then as shown in [ 5 ] , z(w> is determined by the equation