(1977). Graph Theory with Applications. Journal of the Operational Research Society: Vol. 28, Volume 28, issue 1, pp. 237-238.

https://link.springer.com/content/pdf/10.1057%2Fjors.1977.45.pdf

Graph theory with applications

Planar Graphs. Graphs on Higher Surfaces. Degrees. Critical Graphs. The Conjectures of Hadwiger and Hajos. Sparse Graphs. Perfect Graphs. Geometric and Combinatorial Graphs. Algorithms. Constructions. Edge Colorings. Orientations and Flows. Chromatic Polynomials. Hypergraphs. Infinite Chromatic Graphs. Miscellaneous Problems. Indexes.

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Graph Coloring Problems

G. Clar 6n Rule versus Hückel 4n + 2 Rule 3464 H. Hydrocarbons versus Heteroatomic Systems 3465 IV. Hidden Treasures of Kekulé Valence Structures 3466 A. Conjugated Circuits 3467 B. Innate Degree of Freedom 3470 C. Clar Structures 3472 V. Graph Theoretical Approach to Chemical Structure 3473 A. Metric 3473 B. Chemical Graphs 3473 C. Isospectral Graphs 3473 D. Embedded Graphs 3475 E. Partial Ordering 3476 VI. On Enumeration of Benzenoid Hydrocarbons 3477 VII. Kekulé Valence Structures Count 3479 A. Non-branched Cata-condensed Benzenoids 3481 B. Branched Cata-condensed Benzenoids 3482 C. Benzenoid Lattices 3482 D. Peri-condensed Benzenoids 3483 E. Miscellaneous Benzenoids 3484 F. The Approach of Platt 3485 G. Computer Programs for Calculating K 3485 H. Transfer-Matrix Method 3486 I. Use of Recursion Relations 3486 J. Use of Signed Matrices 3487 VIII. Enumeration of Conjugated Circuits 3488 IX. Approximate Approaches versus Ambitious Computations 3490

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Aromaticity of polycyclic conjugated hydrocarbons.

I was asked to write a paper about the major unsolved problems in com-binatorial mathematics. After some thought it seemed better to modify the title to a less pretentious one. Combinatorial mathematics has grown enormously and a genuine survey would have to include not only topics where I have no real competence but also topics about which I n e v er seriously thought, e.g. algorithmic combinatorics, coding theory and matroid theory. There is no doubt that the proof of the conjecture that several simply stated problems have no good algorithm is fundamental and may h a ve i m-portant consequences for many other branches of mathematics, but unfortunately I have no real feeling for these questions and I feel I should leave the subject to those who are more competent. I just heard that Khachiyan 59], has a polynomial algorithm for linear programming. (See also 50].) This is considered a sensational result and during my last stay i n the U.S. many o f m y friends were greatly impressed by i t. First of all I will discuss some problems on set systems. I state only my three favourite problems, but before starting I refer to the survey paper 31].

On the combinatorial problems which I would most like to see solved

In this paper we deal with the substitution decomposition as known for Boolean functions, set systems and relations. It is shown how combinatorial optimization problems, particularly on graphs, partial orders, project networks, (in-) dependence systems and clutters, naturally lead, under weak assumptions, to this kind of decomposition via certain uniqueness results. We then generalize the common features of this type of decomposition from the special cases to a quite general algebraic level, covering infinite cases, and deduce a general Jordan-Holder theorem as well as uniqeness results for the associated composition tree in this general setting. These results are then reinterpreted for the special cases, and the computational aspects of the substitution decomposition are discussed. Throughout, applications to many problems concerning discrete structures are included, as are connections with other approaches in these fields, in particular the split decomposition.

Substitution Decomposition for Discrete Structures and Connections with Combinatorial Optimization

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.

Graph reconstruction—a survey

The group of an X-join of graphs

The P3-graph of a finite simple graph G is the graph whose vertices are the 3-vertex paths of G, with adjacency between two such paths whenever their union is a 4-vertex path or a 3-cycle. In this paper we show that connected fnite simple graphs G and H with isomorphic P3-graphs are either isomorphic or part of three exceptional families. We also characterize all isomorphisms between P3-graphs in terms of the original graphs. © 1997 John Wiley & Sons, Inc. J Graph Theory 26:35–51, 1997

P 3 -isomorphisms for graphs

On montre que si G est un graphe 3-connexe d'ordre 7 au moins, alors tout plus long chemin entre des sommets distincts de G contient au moins deux aretes contractiles. On en deduit que les plus longs cycles de tels graphes contiennent au moins trois aretes contractiles

Longest cycles in 3‐connected graphs contain three contractible edges

It is shown that with one small exception, the 3-connected graphs admitting longest cycles that contain less than four contractible edges of the parent graph are the members of three closely related infinite families © 1993 John Wiley & Sons, Inc

Robert L. Hemminger

Papers

Graph reconstruction—a survey

The group of an X-join of graphs

P 3 -isomorphisms for graphs

Longest cycles in 3‐connected graphs contain three contractible edges

The 3-connected graphs having a longest cycle containing only three contractible edges