In this paper, some results concerning the colorings of graph powers are presented. The notion of helical graphs is introduced. We show that such graphs are hom-universal with respect to high odd-girth graphs whose (2t+1)st power is bounded by a Kneser graph. Also, we consider the problem of existence of homomorphism to odd cycles. We prove that such homomorphism to a (2k+1)cycle exists if and ...

Let n and k be integers with n ≥ k ≥ 0. This paper presents a new class of graphs H(n, k), which contains hypercubes and some well-known graphs, such as Johnson graphs, Kneser graphs and Petersen graph, as its subgraphs. The authors present some results of algebraic and topological properties of H(n, k). For example, H(n, k) is a Cayley graph, the automorphism group of H(n, k) contains a subgro...

We propose a new hierarchical Bayesian n-gram model of natural languages. Our model makes use of a generalization of the commonly used Dirichlet distributions called Pitman-Yor processes which produce power-law distributions more closely resembling those in natural languages. We show that an approximation to the hierarchical Pitman-Yor language model recovers the exact formulation of interpolat...

We consider powers of regular graphs defined by the weak graph product and give a characterization of maximum-size independent sets for a wide family of base graphs which includes, among others, complete graphs, line graphs of regular graphs which contain a perfect matching and Kneser graphs. In many cases this also characterizes the optimal colorings of these products. We show that the indepen...

The local chromatic number of a graph G is the number of colors appearing in the most colorful closed neighborhood of a vertex minimized over all proper colorings of G. We show that two specific topological obstructions that have the same implications for the chromatic number have different implications for the local chromatic number. These two obstructions can be formulated in terms of the hom...

We investigate the empirical behavior of ngram discounts within and across domains. When a language model is trained and evaluated on two corpora from exactly the same domain, discounts are roughly constant, matching the assumptions of modified Kneser-Ney LMs. However, when training and test corpora diverge, the empirical discount grows essentially as a linear function of the n-gram count. We a...

In the multicoloring problem, also known as (a:b)-coloring or b-fold coloring, we are given a graph G and a set of a colors, and the task is to assign a subset of b colors to each vertex of G so that adjacent vertices receive disjoint color subsets. This natural generalization of the classic coloring problem (the b = 1 case) is equivalent to finding a homomorphism to the Kneser graph KGa,b. It ...

Our purpose is to show that complements of line graphs (of graphs) enjoy nice coloring properties. We for all in this class the local and usual chromatic numbers are equal. also prove a sufficient condition number be equal natural upper bound. A consequence latter complete characterization induced subgraphs Kneser graph KG ( n , 2 ) have its number, namely − 2. In addition bound, lower bound pr...

Let V (n, k, s) be the set of k-subsets S of [n] such that for all i, j ∈ S, we have |i−j| ≥ s We define almost s-stable Kneser hypergraph KG ( [n] k )∼ s-stab to be the r-uniform hypergraph whose vertex set is V (n, k, s) and whose edges are the r-uples of disjoint elements of V (n, k, s). With the help of a Zp-Tucker lemma, we prove that, for p prime and for any n ≥ kp, the chromatic number o...