Combinatorics Advances: (329 Mathematics and Its Applications)

On March 28~31, 1994 (Farvardin 8~11, 1373 by Iranian calendar), the Twenty­ fifth Annual Iranian Mathematics Conference (AIMC25) was held at Sharif University of Technology in Tehran, Islamic Republic of Iran. Its sponsors in~ eluded the Iranian Mathematical Society, and the Department of Mathematical Sciences at Sharif University of Technology. Among the keynote speakers were Professor Dr. Andreas Dress and Professor Richard K. Guy. Their plenary lec~ tures on combinatorial themes were complemented by invited and contributed lectures in a Combinatorics Session. This book is a collection of refereed papers, submitted primarily by the participants after the conference. The topics covered are diverse, spanning a wide range of combinatorics and al~ lied areas in discrete mathematics. Perhaps the strength and variety of the pa~ pers here serve as the best indications that combinatorics is advancing quickly, and that the Iranian mathematics community contains very active contributors. We hope that you find the papers mathematically stimulating, and look forward to a long and productive growth of combinatorial mathematics in Iran.

Table of Contents:
1 On a Conjecture of A. Hartman.- 1 Notations and Preliminaries.- 2 The Structure of Trades.- 3 Block Size 3 and Strength 2.- 4 A Possible Approach to the Problem.- 5 The Case of Strength 2.- 6 Some Examples.- 7 Concluding Remarks.- 2 Some Problems in Total Graph Theory.- 1 Introduction and Preliminaries.- 2 Total Ramsey Numbers.- 3 Vertex Reconstructibility of Total Graphs.- 4 Edge Reconstructibility of Total Graphs.- 5 The Spectrum of Total Graphs.- 6 Groups, and Polynomials of Graphs.- 7 Relationships Between some Parameters of G and those of T(G).- 8 Some Generalizations and Applications.- 9 Upper Bounds for x? (G).- 10 Remarks.- 3 Construction Techniques for Mutually Orthogonal Latin Squares.- 1 Background.- 2 History and Small Orders.- 3 Pairwise balanced designs and Greig’s line-flip.- 4 Difference matrices: some direct constructions.- 5 A variant of Wilson’s theorem.- 6 Concluding remarks.- 4 The Spectrum of R-Orthogonal Latin Squares.- 1 Latin squares and r-orthogonality.- 2 Some basic constructions.- 3 Small sides.- 4 A GDD construction.- 5 Intermediate sides.- 5 General Theory of Translation Invariant Systems.- 1 Introduction.- 2 The Model.- 3 A Residuated Semigroup.- 4 Some Basic Questions.- 6 Some Mathematical Problems Arising in Molecular Bioinformatics: The Concept of Bioinformatics.- 1 Introduction.- 2 The concept of sequence space.- 3 The geometry of sequence space.- 4 Cluster analysis.- 5 Split decomposition.- 6 Concluding remark.- 7 An Algorithmic Approach to Tilings.- 8 A New Connection Between Convex Geometry and Threshold Logic.- 1 Introduction.- 2 Algorithmic Enumeration of Nonisomorphic Cut-Complexes and a Generation of Convex Polytopes.- 3 Geometric Properties.- 9 The Unity of Combinatorics.- 1 Introduction.- 2 Langford’s Problem.- 3Skolem’s Problem.- 4 Beatty Sequences.- 5 Penrose Pieces.- 6 Wythoff’s Game.- 7 Triples satisfying x + y = z.- 8 Triples satisfying x + y = 2z.- 9 Coil diagrams.- 10 Squaring the square.- 11 Packing or covering the complete graph.- 12 Hanani’s cyclic Steiner systems.- 13 Perfect difference sets.- 14 Projective planes.- 15 Affine geometries.- 16 Magic squares.- 17 Kirkman’s schoolgirls problem.- 18 Heawood’s map on the torus.- 19 The toroidal thickness of the complete graph.- 20 Nim addition.- 21 Incidence matrices.- 22 Zarankiewicz’s problem.- 23 Error-correcting codes.- 24 Hadamard matrices.- 25 Cyclic Hadamard matrices.- 26 Factoring with quadratic forms.- 27 Projective geometries.- 28 Sphere packing.- 29 Mock Turtles.- 10 Unsolved Problems in Combinatorial Games.- 11 (F, 2)—Rotational Steiner Triple Systems.- 1 Introduction.- 2 Skolem Sequences.- 3 Constructions.- 4 Main Results.- 12 A Simple Polynomial Time Algorithm for a Convex Hull Problem Equivalent to Linear Programming.- 1 Introduction.- 2 The Algorithm.- 13 A Linear—Time Algorithm for Minimum Cost Flow on Undirected One—Trees.- 1 Introduction.- 2 The Algorithm.- 14 An Asymptotic Existence Result for Orthogonal Designs.- 1 Introduction.- 2 Basic Results.- 3 Main Results.- 15 Decomposition of Complete Tripartite Graphs Into 5-Cycles.- 1 Introduction.- 2 Necessary Conditions.- 3 An Application.- 4 Sufficiency of Conditions.- 5 Searching for a decomposition in other cases.- 16 The Nsm of a Graph.- 1 The New Stability Measure of a Graph (NSM).- 2 NSM and Operations on Graphs.- 3 Hamilton Properties of NSM.- 17 Zero-Knowledge Proofs For Independent Set and Dominating Set Problems.- 1 Introduction.- 2 A Zero-Knowledge Proof for Independent set problem.- 3 A Zero-Knowledge Proof for Dominating setproblem.- 18 Exploring the Spectrum of Values of Permanents by Simulated Annealing.- 1 The Permanent.- 2 Upper Bounds and Lower Bounds for the Permanent.- 3 Simulated Annealing.- 4 The Metropolis Algorithm.- 5 Results and Conclusions.- 19 Vector—Weighted Matchings.- 1 Introduction.- 2 Preliminaries.- 3 Preference Matchings.- 4 Preference Poly topes.- 5 The Set of Efficient Solutions.- 20 Directed Quadruple Designs.- 1 Introduction.- 2 Existence of 3-(v, 4, l)DDs.- 3 On the Existence of 3-(v, 4, 2)DDs.- 4 Some Small Cases (for ? = 2).- 21 Bounding Two-Terminal Network Reliability Via Surface Duality.- 1 Introduction.- 2 Definitions.- 3 Results.- 4 Implementation.- 5 Examples.- 22 Defining Sets for Block Designs: An Update.- 1 Introduction.- 2 Some Theoretical Results.- 3 Finding Smallest Defining Sets in Small Designs.- 4 Defining Sets in Some Infinite Classes of Designs.- 23 Open Problems at the Combinatorics Workshop of Aimc25 (Tehran, 1994).