A labeling of a graph is an assignment of a natural number to each vertex of a graph. Graceful labelings are very important types of labelings. The study of graceful labelings is very difficult and little has been shown about such labelings. Vertex-relaxed graceful labelings of graphs are a class of labelings that include graceful labelings, and their study gives an approach to the study of graceful labelings. In this thesis we generalize the congruence approach of Rosa to obtain new criteria for vertex-relaxed graceful labelings of graphs. To do this, we generalize Faulhaber’s Formula, which is a famous result about sums of powers of integers.

In this thesis, we provide an algorithm for constructing strongly eutactic lattices from vertex transitive graphs. We show that such construction produces infinitely many strongly eutactic lattices in arbitrarily large dimensions. We demonstrate our algorithm on the example of the famous Petersen graph using Maple computer algebra system. We also discuss some additional examples of strongly eutactic lattices obtained from notable vertex transitive graphs. Further, we study the properties of the lattices generated by product graphs, complement graphs, and line graphs of vertex transitive graphs. This thesis is based on the research paper written by the author jointly with L. Fukshansky, D. Needell and J. Park.

The associahedron has appeared in numerous contexts throughout the field of mathematics. By representing the associahedron as a poset of tubings, Michael Carr and Satyan L. Devadoss were able to create a gener- alized version of the associahedron in the graph-associahedron. We seek to create an alternative generalization of the associahedron by considering a particle-collision model. By extending this model to what we dub the 2- associahedron, we seek to further understand the space of generalizations of the associahedron.

L'etude et la comprehension de phenomenes naturels qu'il<br />semble difficile de predire, tels les<br />tremblements de terre et les raz de maree intriguent depuis quelques<br />temps un certain nombre de physiciens. En effet, il semble que les<br />modeles classiques bases sur des fonctions d'etat continues<br />peuvent difficilement expliquer les phenomenes observes.<br /><br />En 1987, Bak, Tang et Wiesenfeld introduisent un modele base<br /> sur un automate particulier dont l'etude experimentale montre<br /> des caracteristiques proches de celles observees pour des<br /> tremblements de terre. Cet automate est appele automate du tas de sable.<br /><br /> En 1990, Dhar, Ruelle, Sen et<br />Verma etudient les proprietes mathematiques<br />de l'automate du tas de sable. Cet article jette les bases d'une théorie algebrique et combinatoire des<br />etats critiques du systeme en montrant que ceux-ci forment un<br />groupe abelien fini.<br /><br />Cette these porte essentiellement sur l'etude de ce groupe d'un<br />point de vue algorithmique, combinatoire et algebrique. Nous<br />etudions dans un premier temps la complexite de l'operateur de<br />groupe. Puis nous etudions le groupe sur quelques familles de<br />graphes connues avant de montrer que le groupe d'un graphe planaire<br />est isomorphe au groupe de chacun de ses duaux geometriques.<br /><br />Nous montrons comment associer à un groupe abelien fini un<br />idéal de polynomes et dans le cas du groupe du Tas de Sable, nous<br />donnerons une caracterisation de l'operateur de groupe en terme de<br />reduction de polynome.

Cette thèse traite de phénomènes de seuils et de modèles d'urnes, en adoptant le point de vue de la combinatoire analytique. On traite ici trois problèmes qui illustrent cette approche: la transition de phase du problème k-sat, les modèles d'urnes triangulaires de Polya-Eggenberger et le modèle de duel de Ok Corral. La transition de phase du problème k-sat se manifeste par le fait la densité d'une formule caractérise de manière presque sûre sa satisfaisabilité. Nos travaux visent à mettre en évidence une partie de ce phénomène et se relient à un modèle d'urne à jets. Le modèle d'urne de Polya-Eggenberger utilise une urne contenant des boules de diverses couleurs, soumises à des règles de pioches et de substitutions. En utilisant une technique de Flajolet-Gabarro-Pekari, nous déterminons la distribution limite de la composition des modèles dits triangulaires. Le modèle de duel de Ok Corral intervient dans une problématique plus générale de Lanchester de gestion des conflits, selon laquelle on cherche à prédire l'issue de duels entre plusieurs forces armées, en environnement aléatoire. Nous utilisons un lien pré-établi entre ce modèle et une urne de type Polya-Eggenberger pour donner de nouvelles expressions des probabilités du modèle et raffiner les résultats récents de Kingman.

Many recognition systems use constrained search to locate objects in cluttered environments. Earlier analysis showed that the expected search is quadratic in the number of model and data features, if all the data comes from one object, but is exponential when spurious data is included. To overcome this, many methods terminate search once an interpretation that is "good enough" is found. We formally examine the combinatorics of this, showing that correct termination procedures dramatically reduce search. We provide conditions on the object model and the scene clutter such that the expected search is quartic. These results are shown to agree with empirical data for cluttered object recognition.

We provide a simple and improved security analysis of PMAC, a Parallelizable MAC (Message Authentication Code) defined over arbitrary messages. A similar kind of result was shown by Bellare, Pietrzak and Rogaway at Crypto 2005, where they have provided an improved bound for CBC (Cipher Block Chaining) MAC, which was introduced by Bellare, Killan and Rogaway at Crypto 1994. Our analysis idea is much more simpler to understand and is borrowed from the work by Nandi for proving Indistinguishability at Indocrypt 2005 and work by Bernstein. It shows that the advantage for any distinguishing attack for n-bit PMAC based on a random function is bounded by O(σq / 2^n), where σ is the total number of blocks in all q queries made by the attacker. In the original paper by Black and Rogaway at Eurocrypt 2002 where PMAC was introduced, the bound is O(σ^2 / 2^n). We also compute the collision probability of CBC MAC for suitably chosen messages. We show that the probability is Ω( lq^2 / N) where l is the number of message blocks, N is the size of the domain and q is the total number of queries. For random oracles the probability is O(q^2 / N). This improved collision probability will help us to have an efficient distinguishing attack and MAC-forgery attack. We also show that the collision probability for PMAC is Ω(q^2 / N) (strictly greater than the birthday bound). We have used a purely combinatorial approach to obtain this bound. Similar analysis can be made for other CBC MAC extensions like XCBC, TMAC and OMAC.

One question relating to partially ordered sets (posets) is that of partitioning or dividing the poset's elements into the fewest number of chains that span the poset. In 1950, Dilworth established that the width of the poset - the size of the largest set composed only of incomparable elements - is the minimum number of chains needed to partition that poset. Such a bound in on-line partitioning has been harder to establish, and work has evalutated classes of posets based on their width. This paper reviews the theorems that established val(2)=5 and illustrates them with examples. It also covers some of the work on establishing bounds for on-line partitioning with the Greedy Algorithm. The paper concludes by contributing a bound on incomparable elements in graded, (t+t)-free, finite width posets.

The Lovasz theta function and the associated convex sets known as theta bodies are fundamental objects in combinatorial and semidefinite optimization. They are accompanied by a rich duality theory and deep connections to the geometric concept of orthonormal representations of graphs. In this thesis, we investigate several ramifications of the theory underlying these objects, including those arising from the illuminating viewpoint of duality. We study some optimization problems over unit-distance representations of graphs, which are intimately related to the Lovasz theta function and orthonormal representations. We also strengthen some known results about dual descriptions of theta bodies and their variants. Our main goal throughout the thesis is to lay some of the foundations for using semidefinite optimization and convex analysis in a way analogous to how polyhedral combinatorics has been using linear optimization to prove min-max theorems. A unit-distance representation of a graph $G$ maps its nodes to some Euclidean space so that adjacent nodes are sent to pairs of points at distance one. The hypersphere number of $G$, denoted by $t(G)$, is the (square of the) minimum radius of a hypersphere that contains a unit-distance representation of $G$. Lovasz proved a min-max relation describing $t(G)$ as a function of $\vartheta(\overline{G})$, the theta number of the complement of $G$. This relation provides a dictionary between unit-distance representations in hyperspheres and orthonormal representations, which we exploit in a number of ways: we develop a weighted generalization of $t(G)$, parallel to the weighted version of $\vartheta$; we prove that $t(G)$ is equal to the (square of the) minimum radius of an Euclidean ball that contains a unit-distance representation of $G$; we abstract some properties of $\vartheta$ that yield the famous Sandwich Theorem and use them to define another weighted generalization of $t(G)$, called ellipsoidal number of $G$, where the unit-distance representation of $G$ is required to be in an ellipsoid of a given shape with minimum volume. We determine an analytic formula for the ellipsoidal number of the complete graph on $n$ nodes whenever there exists a Hadamard matrix of order $n$. We then study several duality aspects of the description of the theta body $\operatorname{TH}(G)$. For a graph $G$, the convex corner $\operatorname{TH}(G)$ is known to be the projection of a certain convex set, denoted by $\widehat{\operatorname{TH}}(G)$, which lies in a much higher-dimensional matrix space. We prove that the vertices of $\widehat{\operatorname{TH}}(G)$ are precisely the symmetric tensors of incidence vectors of stable sets in $G$, thus broadly generalizing previous results about vertices of the elliptope due to Laurent and Poljak from 1995. Along the way, we also identify all the vertices of several variants of $\widehat{\operatorname{TH}}(G)$ and of the elliptope. Next we introduce an axiomatic framework for studying generalized theta bodies, based on the concept of diagonally scaling invariant cones, which allows us to prove in a unified way several characterizations of $\vartheta$ and the variants $\vartheta'$ and $\vartheta^+$, introduced independently by Schrijver, and by McEliece, Rodemich, and Rumsey in the late 1970's, and by Szegedy in 1994. The beautiful duality equation which states that the antiblocker of $\operatorname{TH}(G)$ is $\operatorname{TH}(\overline{G})$ is extended to this setting. The framework allows us to treat the stable set polytope and its classical polyhedral relaxations as generalized theta bodies, using the completely positive cone and its dual, and it allows us to derive a (weighted generalization of a) copositive formulation for the fractional chromatic number due to Dukanovic and Rendl in 2010 from a completely positive formulation for the stability number due to de Klerk and Pasechnik in 2002. Finally, we study a non-convex constraint for semidefinite programs (SDPs) that may be regarded as analogous to the usual integrality constraint for linear programs. When applied to certain classical SDPs, it specializes to the standard rank-one constraint. More importantly, the non-convex constraint also applies to the dual SDP, and for a certain SDP formulation of $\vartheta$, the modified dual yields precisely the clique covering number. This opens the way to study some exactness properties of SDP relaxations for combinatorial optimization problems akin to the corresponding classical notions from polyhedral combinatorics, as well as approximation algorithms based on SDP relaxations.

We consider the problem of counting <i>transitive factorizations</i> of permutations; that is, we study tuples (&sigma;<i>r</i>,. . . ,&sigma;1) of permutations on {1,. . . ,<i>n</i>} such that (1) the product &sigma;<i>r</i>. . . &sigma;1 is equal to a given target permutation &pi;, and (2) the group generated by the factors &sigma;<i>i</i> acts transitively on {1,. . . ,<i>n</i>}. This problem is widely known as the <i>Hurwitz Enumeration Problem</i>, since an encoding due to Hurwitz shows it to be equivalent to the enumeration of connected branched coverings of the sphere by a surface of given genus with specified branching. Much of our work concerns the enumeration of transitive factorizations of permutations into a minimal number of transposition factors. This problem has received considerable attention, and a formula for the number <i>c</i>(&pi;) of such factorizations of an arbitrary permutation &pi; has been derived through various means. The formula is remarkably simple, being a product of well-known combinatorial numbers, but no bijective proof of it is known except in the special case where &pi; is a full cycle. A major goal of this thesis is to provide further combinatorial rationale for this formula. We begin by introducing an encoding of factorizations (into transpositions) as edge-labelled maps. Our central result is a bijection that allows trees to be "pruned" from such maps. This is shown to explain the appearance of factors of the form <i>k^k</i> in the aforementioned formula for <i>c</i>(&pi;). It also has the effect of shifting focus to the combinatorics of smooth maps (<i>i. e. </i> maps without vertices of degree one). By providing decompositions for certain smooth planar maps, we are able to give combinatorial evaluations of <i>c</i>(&pi;) when &pi; is composed of up to three cycles. Many of these results are generalized to factorizations in which the factors are cycles of any length. We also investigate the <i>Double Hurwitz Problem</i>, which calls for the enumeration of factorizations whose leftmost factor is of specified cycle type, and whose remaining factors are transpositions. Finally, we extend our methods to the enumeration of factorizations up to an equivalence relation induced by possible commutations between adjacent factors.