The Strong Perfect Graph Conjecture: 40 years of Attempts, and its Resolution

International audienceThe Strong Perfect Graph Conjecture (SPGC) was certainly one of the most challenging conjectures in graph theory. During more than four decades, numerous attempts were made to solve it, by combinatorial methods, by linear algebraic methods, or by polyhedral methods. The first of these three approaches yielded the first (and to date only) proof of the SPGC; the other two remain promising to consider in attempting an alternative proof. This paper is an unbalanced survey of the attempts to solve the SPGC; unbalanced, because (1) we devote a signicant part of it to the 'primitive graphs and structural faults' paradigm which led to the Strong Perfect Graph Theorem (SPGT); (2) we briefly present the other "direct" attempts, that is, the ones for which results exist showing one (possible) way to the proof; (3) we ignore entirely the "indirect" approaches whose aim was to get more information about the properties and structure of perfect graphs, without a direct impact on the SPGC. Our aim in this paper is to trace the path that led to the proof of the SPGT as completely as possible. Of course, this implies large overlaps with the recent book on perfect graphs [J.L. Ramirez-Alfonsin and B.A. Reed, eds., Perfect Graphs (Wiley & Sons, 2001).], but it also implies a deeper analysis (with additional results) and another viewpoint on the topic

Edge-colourings of graphs

All the results in this thesis are concerned with the classification of graphs by their chromatic class. We first extend earlier results of Fiorini and others to give a complete list of critical graphs of order at most ten. We give conditions for extending the edge-colouring of a nearly complete subgraph to the whole graph and use this result to prove a special case of Vizing's conjecture. We also use other methods to solve further cases of this conjecture. A major part of the thesis classifies those graphs with at most 4 vertices of maximum degree and this work is generalised to graphs with r vertices of maximum degree. We also completely classify all regular graphs G with degree at least 6/7|V(G)|. Finally we give some examples of even order critical graphs and introduce the concept of a supersnark

Structural and Topological Graph Theory and Well-Quasi-Ordering

Στη σειρά εργασιών Ελασσόνων Γραφημάτων, οι Neil Robertson και Paul Seymour μεταξύ άλλων σπουδαίων αποτελεσμάτων, απέδειξαν την εικασία του Wagner που σήμερα είναι γνωστή ως το Θεώρημα των Robertson και Seymour. Σε κάθε τους βήμα προς την συναγωγή της τελικής απόδειξης της εικασίας, κάθε ειδική περίπτωση αυτής που αποδείκνυαν ήταν συνέπεια ενός "δομικού θεωρήματος" το οποίο σε γενικές γραμμές ισχυριζόταν ότι ικανοποιητικά γενικά γραφήματα περιέχουν ως ελάσσονα γραφήματα ή άλλες δομές που είναι χρήσιμα για την απόδειξη, ή ισοδύναμα, ότι η δομή των γραφημάτων τα οποία δεν περιέχουν ένα χρήσιμο για την απόδειξη γράφημα ως έλασσον είναι κατά κάποιο τρόπο περιορισμένη συνάγοντας έτσι και πάλι μια χρήσιμη πληροφορία για την απόδειξη. Στην παρούσα εργασία, παρουσιάζουμε -σχετικά μικρές- αποδείξεις διαφόρων ειδικών περιπτώσεων του Θεωρήματος των Robertson και Seymour, αναδεικνύοντας με αυτό τον τρόπο την αλληλεπίδραση της δομικής θεωρίας γραφημάτων με την θεωρία των καλών-σχεδόν-διατάξεων. Παρουσιάζουμε ακόμα την ίσως πιο ενδιαφέρουσα ειδική περίπτωση του Θεωρήματος των Robertson και Seymour, η οποία ισχυρίζεται ότι η εμβαπτισιμότητα σε κάθε συγκεκριμένη επιφάνεια δύναται να χαρακτηριστεί μέσω της απαγόρευσης πεπερασμένων το πλήθος γραφημάτων ως ελάσσονα. Το τελευταίο αποτέλεσμα συνάγεται ως ένα αποτέλεσμα της θεωρίας των καλών-σχεδόν-διατάξεων αναδεικνύοντας με αυτό τον τρόπο την αλληλεπίδρασή της με την τοπολογική θεωρία γραφημάτων. Τέλος, σταχυολογούμε αποτελέσματα αναφορικά με την καλή-σχεδόν-διάταξη κλάσεων γραφημάτων από άλλες -πέραν της σχέσης έλασσον- σχέσεις γραφημάτων.In their Graph Minors series, Neil Robertson and Paul Seymour among other great results proved Wagner's conjecture which is today known as the Robertson and Seymour's theorem. In every step along their way to the final proof, each special case of the conjecture which they were proving was a consequence of a "structure theorem", that sufficiently general graphs contain minors or other sub-objects that are useful for the proof - or equivalently, that graphs that do not contain a useful minor have a certain restricted structure, deducing that way also a useful information for the proof. The main object of this thesis is the presentation of -relatively short- proofs of several Robertson and Seymour's theorem's special cases, illustrating by this way the interplay between structural graph theory and graphs' well-quasi-ordering. We present also the proof of the perhaps most important special case of the Robertson and Seymour's theorem which states that embeddability in any fixed surface can be characterized by forbidding finitely many minors. The later result is deduced as a well-quasi-ordering result, indicating by this way the interplay among topological graph theory and well-quasi-ordering theory. Finally, we survey results regarding the well-quasi-ordering of graphs by other than the minor graphs' relations

Gauge Theory of Elementary Particle Physics

The aim of this book is to provide student and researcher with a practical introduction to some of the principal ideas in gauge theories and their applications to elementary particle physics. Elementary particle physics has made remarkable progress. We have a comprehensive theory of particle interactions. One can argue that it gives a complete and correct description of all non-gravitational physics. This theory is based on the principle of gauge symmetry. Strong, weak, and electromagnetic interactions are all gauge interactions. A knowledge of gauge theory is essential for anyone interested in modern high energy physics. Regardless of the ultimate correctness of every detail of this theory, it is the framework within which new theoretical and experimental advances will be interpreted in the foreseeable future

Gauge Theory of Elementary Particle Physics

The aim of this book is to provide student and researcher with a practical introduction to some of the principal ideas in gauge theories and their applications to elementary particle physics. Elementary particle physics has made remarkable progress. We have a comprehensive theory of particle interactions. One can argue that it gives a complete and correct description of all non-gravitational physics. This theory is based on the principle of gauge symmetry. Strong, weak, and electromagnetic interactions are all gauge interactions. A knowledge of gauge theory is essential for anyone interested in modern high energy physics. Regardless of the ultimate correctness of every detail of this theory, it is the framework within which new theoretical and experimental advances will be interpreted in the foreseeable future