Combinatorics Advances

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).