Entropy of random coverings and 4D quantum gravity

We discuss the counting of minimal geodesic ball coverings of $n$-dimensional riemannian manifolds of bounded geometry, fixed Euler characteristic and Reidemeister torsion in a given representation of the fundamental group. This counting bears relevance to the analysis of the continuum limit of discrete models of quantum gravity. We establish the conditions under which the number of coverings grows exponentially with the volume, thus allowing for the search of a continuum limit of the corresponding discretized models. The resulting entropy estimates depend on representations of the fundamental group of the manifold through the corresponding Reidemeister torsion. We discuss the sum over inequivalent representations both in the two-dimensional and in the four-dimensional case. Explicit entropy functions as well as significant bounds on the associated critical exponents are obtained in both cases.Comment: 54 pages, latex, no figure

On n-fold L(j,k)-and circular L(j,k)-labelings of graphs

AbstractWe initiate research on the multiple distance 2 labeling of graphs in this paper.Let n,j,k be positive integers. An n-fold L(j,k)-labeling of a graph G is an assignment f of sets of nonnegative integers of order n to the vertices of G such that, for any two vertices u,v and any two integers a∈f(u), b∈f(v), |a−b|≥j if uv∈E(G), and |a−b|≥k if u and v are distance 2 apart. The span of f is the absolute difference between the maximum and minimum integers used by f. The n-fold L(j,k)-labeling number of G is the minimum span over all n-fold L(j,k)-labelings of G.Let n,j,k and m be positive integers. An n-fold circular m-L(j,k)-labeling of a graph G is an assignment f of subsets of {0,1,…,m−1} of order n to the vertices of G such that, for any two vertices u,v and any two integers a∈f(u), b∈f(v), min{|a−b|,m−|a−b|}≥j if uv∈E(G), and min{|a−b|,m−|a−b|}≥k if u and v are distance 2 apart. The minimum m such that G has an n-fold circular m-L(j,k)-labeling is called the n-fold circular L(j,k)-labeling number of G.We investigate the basic properties of n-fold L(j,k)-labelings and circular L(j,k)-labelings of graphs. The n-fold circular L(j,k)-labeling numbers of trees, and the hexagonal and p-dimensional square lattices are determined. The upper and lower bounds for the n-fold L(j,k)-labeling numbers of trees are obtained. In most cases, these bounds are attainable. In particular, when k=1 both the lower and the upper bounds are sharp. In many cases, the n-fold L(j,k)-labeling numbers of the hexagonal and p-dimensional square lattices are determined. In other cases, upper and lower bounds are provided. In particular, we obtain the exact values of the n-fold L(j,1)-labeling numbers of the hexagonal and p-dimensional square lattices

From Bruhat intervals to intersection lattices and a conjecture of Postnikov

We prove the conjecture of A. Postnikov that (A) the number of regions in the inversion hyperplane arrangement associated with a permutation w\in \Sn is at most the number of elements below $w$ in the Bruhat order, and (B) that equality holds if and only if $w$ avoids the patterns 4231, 35142, 42513 and 351624. Furthermore, assertion (A) is extended to all finite reflection groups. A byproduct of this result and its proof is a set of inequalities relating Betti numbers of complexified inversion arrangements to Betti numbers of closed Schubert cells. Another consequence is a simple combinatorial interpretation of the chromatic polynomial of the inversion graph of a permutation which avoids the above patterns.Comment: 24 page

More relations between λ\lambda-labeling and Hamiltonian paths with emphasis on line graph of bipartite multigraphs

This paper deals with the $\lambda$-labeling and $L(2,1)$-coloring of simple graphs. A $\lambda$-labeling of a graph $G$ is any labeling of the vertices of $G$ with different labels such that any two adjacent vertices receive labels which differ at least two. Also an $L(2,1)$-coloring of $G$ is any labeling of the vertices of $G$ such that any two adjacent vertices receive labels which differ at least two and any two vertices with distance two receive distinct labels. Assume that a partial $\lambda$-labeling $f$ is given in a graph $G$. A general question is whether $f$ can be extended to a $\lambda$-labeling of $G$. We show that the extension is feasible if and only if a Hamiltonian path consistent with some distance constraints exists in the complement of $G$. Then we consider line graph of bipartite multigraphs and determine the minimum number of labels in $L(2,1)$-coloring and $\lambda$-labeling of these graphs. In fact we obtain easily computable formulas for the path covering number and the maximum path of the complement of these graphs. We obtain a polynomial time algorithm which generates all Hamiltonian paths in the related graphs. A special case is the Cartesian product graph $K_n\Box K_n$ and the generation of $\lambda$-squares.Comment: 20 pages, 7 figures, accepted pape

On the Structure of Graphs with Non-Surjective L(2,1)-Labelings

For a graph G, an L(2,1)-labeling of G with span k is a mapping L \right arrow \{0, 1, 2, \ldots, k\} such that adjacent vertices are assigned integers which differ by at least 2, vertices at distance two are assigned integers which differ by at least 1, and the image of L includes 0 and k. The minimum span over all L(2,1)-labelings of G is denoted $\lambda(G)$, and each L(2,1)-labeling with span $\lambda(G)$ is called a $\lambda$-labeling. For $h \in \{1, \ldots, k-1\}$, h is a hole of Lif and only if h is not in the image of L. The minimum number of holes over all $\lambda$-labelings is denoted $\rho(G)$, and the minimum k for which there exists a surjective L(2,1)-labeling onto {0,1, ..., k} is denoted $\mu(G)$. This paper extends the work of Fishburn and Roberts on $\rho$ and $\mu$ through the investigation of an equivalence relation on the set of $\lambda$-labelings with $\rho$ holes. In particular, we establish that $\rho \leq \Delta$. We analyze the structure of those graphs for which $\rho \in \{ \Delta-1, \Delta \}$, and we show that $\mu = \lambda+ 1$ whenever $\lambda$ is less than the order of the graph. Finally, we give constructions of connected graphs with $\rho = \Delta$ and order $t(\Delta + 1)$, $1 \leq t \leq \Delta$

Entropy estimates for Simplicial Quantum Gravity

Through techniques of controlled topology we determine the entropy function characterizing the distribution of combinatorially inequivalent metric ball coverings of n-dimensional manifolds of bounded geometry for every n ≥ 2. Such functions control the asymptotic distribution of dynamical triangulations of the corresponding n-dimensional (pseudo)manifolds M of bounded geometry. They have an exponential leading behavior determined by the Reidemeister-Franz torsion associated with orthogonal representations of the fundamental group of the manifold. The subleading terms are instead controlled by the Euler characteristic of M. Such results are either consistent with the known asymptotics of dynamically triangulated two-dimensional surfaces, or with the numerical evidence supporting an exponential leading behavior for the number of inequivalent dynamical triangulations on three- and four-dimensional manifolds

L(2,1)-labelling of graphs

International audienceAn $L(2,1)$-labelling of a graph is a function $f$ from the vertex set to the positive integers such that $|f(x)-f(y)|\geq 2$ if $dist(x,y)=1$ and $|f(x)-f(y)|\geq 1$ if $dist(x,y)=2$, where $dist(u,v)$ is the distance between the two vertices~$u$ and~$v$ in the graph $G$. The \emph{span} of an $L(2,1)$-labelling $f$ is the difference between the largest and the smallest labels used by $f$ plus $1$. In 1992, Griggs and Yeh conjectured that every graph with maximum degree $\Delta\geq 2$ has an $L(2,1)$-labelling with span at most $\Delta^2+1$. We settle this conjecture for $\Delta$ sufficiently large.Un $L(2,1)$-étiquettage d'un graphe est une fonction $f$ de l'ensemble des sommets vers les entiers positifs telle que $|f(x)-f(y)|\geq 2$ si $dist(x,y)=1$ et $|f(x)-f(y)|\geq 1$ si $dist(x,y)=2$, où $dist(u,v)$ est la distance entre les sommets~$u$ et~$v$ dans le graphe $G$. Le \emph{span} d'un $L(2,1)$-étiquettage $f$ est la différence entre la plus grande et la plus petite étiquette utilisée par $f$ plus $1$. En 1992, Griggs et Yeh ont conjecturé que tout graphe de degré maximum $\Delta\geq 2$ a un $L(2,1)$-étiquettage de span au plus $\Delta^2+1$. Nous confirmons cette conjecture pour $\Delta$ suffisamment grand

L(2,1)-labelling of graphs

International audienceAn $L(2,1)$-labelling of a graph is a function $f$ from the vertex set to the positive integers such that $|f(x)-f(y)|\geq 2$ if $dist(x,y)=1$ and $|f(x)-f(y)|\geq 1$ if $dist(x,y)=2$, where $dist(u,v)$ is the distance between the two vertices~$u$ and~$v$ in the graph $G$. The \emph{span} of an $L(2,1)$-labelling $f$ is the difference between the largest and the smallest labels used by $f$ plus $1$. In 1992, Griggs and Yeh conjectured that every graph with maximum degree $\Delta\geq 2$ has an $L(2,1)$-labelling with span at most $\Delta^2+1$. We settle this conjecture for $\Delta$ sufficiently large.Un $L(2,1)$-étiquettage d'un graphe est une fonction $f$ de l'ensemble des sommets vers les entiers positifs telle que $|f(x)-f(y)|\geq 2$ si $dist(x,y)=1$ et $|f(x)-f(y)|\geq 1$ si $dist(x,y)=2$, où $dist(u,v)$ est la distance entre les sommets~$u$ et~$v$ dans le graphe $G$. Le \emph{span} d'un $L(2,1)$-étiquettage $f$ est la différence entre la plus grande et la plus petite étiquette utilisée par $f$ plus $1$. En 1992, Griggs et Yeh ont conjecturé que tout graphe de degré maximum $\Delta\geq 2$ a un $L(2,1)$-étiquettage de span au plus $\Delta^2+1$. Nous confirmons cette conjecture pour $\Delta$ suffisamment grand