Edge choosability and total choosability of planar graphs with no 3-cycles adjacent 4-cycles

Two cycles are said to be adjacent if they share a common edge. Let G be a planar graph without triangles adjacent 4-cycles. We prove that if Δ(G)≥6, and and if Δ(G)≥8, where and denote the list edge chromatic number and list total chromatic number of G, respectively.     

Higher order log-concavity in Euler’s difference table

For 0≤k≤n, let be the entries in Euler’s difference table and let . Dumont and Randrianarivony showed equals the number of permutations on [n] whose fixed points are contained in {1,2,…,k}. Rakotondrajao found a combinatorial interpretation of the number in terms of k-fixed-points-permutations of [n]. We show that for any n≥1, the sequence is essentially 2-log-concave and reverse ultra log-concave.     

A new characterization of projections of quadrics in finite projective spaces of even characteristic

We will classify, up to linear representations, all geometries fully embedded in an affine space with the property that for every antiflag {p,L} of the geometry there are either 0, α, or q lines through p intersecting L. An example of such a geometry with α=2 is the following well known geometry . Let Q_n+1 be a nonsingular quadric in a finite projective space , n≥3, q even. We project Q_n+1 from a point r∉Q_n+1, distinct from its nucleus if n+1 is even, on a hyperplane not through r. This yields a partial linear space whose points are the points p of , such that the line 〈p,r〉 is a secant to Q_n+1, and whose lines are the lines of which contain q such points. This geometry is fully embedded in an affine subspace of and satisfies the antiflag property mentioned. As a result of our classification theorem we will give a new characterization theorem of this geometry.     

Logarithmic Sobolev Inequalities for Pinned Loop Groups

LetGbe a connected compact type Lie group equipped with anAd_G-invariant inner product on the Lie algebra ofG. Given this data there is a well known left invariant “H¹-Riemannian structure” on =(G)—the infinite dimensional group of continuous based loops inG. Using this Riemannian structure, we define and construct a “heat kernel”ν_T(g₀, ·) associated to the Laplace–Beltrami operator on (G). HereT>0,g₀∈(G), andν_T(g₀,·) is a certain probability measure on (G). For fixedg₀∈(G) andT>0, we use the measureν_T(g₀,·) and the Riemannian structure on (G) to construct a “classical” pre-Dirichlet form. The main theorem of this paper asserts that this pre-Dirichlet form admits a logarithmic Sobolev inequality.     

Mutually orthogonal equitable Latin rectangles

Let ab=n². We define an equitable Latin rectangle as an a×b matrix on a set of n symbols where each symbol appears either or times in each row of the matrix and either or times in each column of the matrix. Two equitable Latin rectangles are orthogonal in the usual way. Denote a set of ka×b mutually orthogonal equitable Latin rectangles as a k– MOELR (a,b;n). When a≠9,18,36, or 100, then we show that the maximum number of k– MOELR (a,b;n)≥3 for all possible values of (a,b).     

Problems arising from jackknifing the estimate of a Kaplan-Meier integral

Stute and Wang (1994) considered the problem of estimating the integral S^θ = ∫ θ dF, based on a possibly censored sample from a distribution F, where θ is an F-integrable function. They proposed a Kaplan-Meier integral to approximate S^θ and derived an explicit formula for the delete-1 jackknife estimate . differs from only when the largest observation, X_(n), is not censored (δ_(n) = 1 and next-to-the-largest observation, X_(n-1), is censored (δ_(n-1) = 0). In this note, it will pointed out that when X_(n) is censored is based on a defective distribution, and therefore can badly underestimate . We derive an explicit formula for the delete-2 jackknife estimate . However, on comparing the expressions of and , their difference is negligible. To improve the performance of and , we propose a modified estimator according to Efron (1980). Simulation results demonstrate that is much less biased than and and .     

Adjoint polynomials of bridge–path and bridge–cycle graphs and Chebyshev polynomials

The chromatic polynomial of a simple graph G with n>0 vertices is a polynomial of degree n, where α_k(G) is the number of k-independent partitions of G for all k. The adjoint polynomial of G is defined to be , where is the complement of G. We find explicit formulas for the adjoint polynomials of the bridge–path and bridge–cycle graphs. Consequence, we find the zeros of the adjoint polynomials of several families of graphs.     

Random induced subgraphs of Cayley graphs induced by transpositions

In this paper we study random induced subgraphs of Cayley graphs of the symmetric group induced by an arbitrary minimal generating set of transpositions. A random induced subgraph of this Cayley graph is obtained by selecting permutations with independent probability, λ_n. Our main result is that for any minimal generating set of transpositions, for probabilities where and δ>0, a random induced subgraph has a.s. a unique largest component of size . Here x(?_n) is the survival probability of a Poisson branching process with parameter λ=1+?_n.     

Closure operation for even factors on claw-free graphs

Ryjá?ek (1997) [6] defined a powerful closure operation on claw-free graphs G. Very recently, Ryjá?ek et al. (2010) [8] have developed the closure operation on claw-free graphs which preserves the (non)-existence of a 2-factor. In this paper, we introduce a closure operation on claw-free graphs that generalizes the above two closure operations. The closure of a graph is unique determined and the closure turns a claw-free graph into the line graph of a graph containing no cycle of length at most 5 and no cycles of length 6 satisfying a certain condition and no induced subgraph being isomorphic to the unique tree with a degree sequence 111133. We show that these closure operations on claw-free graphs all preserve the minimum number of components of an even factor. In particular, we show that a claw-free graph G has an even factor with at most k components if and only if (, respectively) has an even factor with at most k components. However, the closure operation does not preserve the (non)-existence of a 2-factor.     

Backward Shift Invariant Operator Ranges

Results on first order Ext groups for Hilbert modules over the disk algebra are used to study certain backward shift invariant operator ranges, namely de Branges–Rovnyak spaces and a more general class called (W; B) spaces. Necessary and sufficient conditions are given for the groups Ext¹_A()(, (W; B)) to vanish whereis thedualof the vector-valued Hardy module, H². One condition involves an extension problem for the Hankel operator with symbolB,Γ_B, but viewed as a module map from H²into (W; B). The group Ext¹_A()(, (W; B))=(0) precisely whenΓ_Bextends to a module map from L²into (W; B) and this in turn is equivalent to the injectivity of (W; B) in the category of contractive HilbertA()-modules. This result applied to the de Branges–Rovnyak spaces yields a connection between the extension problem for the HankelΓ_B and the operator corona problem.     

Coquet-type formulas for the rarefied weighted Thue–Morse sequence

Newman proved for the classical Thue–Morse sequence, ((−1)^s(n))_n≥0, that for all N∈N with real constants satisfying c₂>c₁>0 and λ=log3/log4. Coquet improved this result and deduced , where F(x) is a nowhere-differentiable, continuous function with period 1 and η(N)∈{−1,0,1}. In this paper we obtain for the weighted version of the Thue–Morse sequence that for the sum a Coquet-type formula exists for every r∈{0,1,2} if and only if the sequence of weights is eventually periodic. From the specific Coquet-type formulas we derive parts of the weak Newman-type results that were recently obtained by Larcher and Zellinger.     

Forbidden configurations and repeated induction

For a given k×? matrix F, we say a matrix A has no configurationF if no k×? submatrix of A is a row and column permutation of F. We say a matrix is simple if it is a (0,1)-matrix with no repeated columns. We define as the maximum number of columns in an m-rowed simple matrix which has no configuration F. A fundamental result of Sauer, Perles and Shelah, and Vapnik and Chervonenkis determines exactly, where K_k denotes the k×2^k simple matrix. We extend this in several ways. For two matrices G,H on the same number of rows, let [G∣H] denote the concatenation of G and H. Our first two sets of results are exact bounds that find some matrices B,C where and . Our final result provides asymptotic boundary cases; namely matrices F for which is O(m^p) yet for any choice of column α not in F, we have is Ω(m^p+1). This is evidence for a conjecture of Anstee and Sali. The proof techniques in this paper are dominated by repeated use of the standard induction employed in forbidden configurations. Analysis of base cases tends to dominate the arguments. For a k-rowed (0,1)-matrix F, we also consider a function which is the minimum number of columns in an m-rowed simple matrix for which each k-set of rows contains F as a configuration.     

On the eccentric connectivity index of a graph

If G is a connected graph with vertex set V, then the eccentric connectivity index of G, ξ^C(G), is defined as where is the degree of a vertex v and is its eccentricity. We obtain an exact lower bound on ξ^C(G) in terms of order, and show that this bound is sharp. An asymptotically sharp upper bound is also derived. In addition, for trees of given order, when the diameter is also prescribed, precise upper and lower bounds are provided.     

The Real Tree

In this note, it is shown that there exists a natural metric on the set[formula]of bounded subsets of containing their infimum which endowsTwith the structure of an -tree so that, for everyt∈T, the “number” of connected components ofT\{t} coincides with the cardinality #() of the set of subsets of . In addition, the set of ends ofTis explicitly determined, and various further features ofTare discussed, too.     

Linear choosability of sparse graphs

A linear coloring is a proper coloring such that each pair of color classes induces a union of disjoint paths. We study the linear list chromatic number, denoted , of sparse graphs. The maximum average degree of a graph G, denoted mad(G), is the maximum of the average degrees of all subgraphs of G. It is clear that any graph G with maximum degree Δ(G) satisfies . In this paper, we prove the following results: (1) if and Δ(G)≥3, then , and we give an infinite family of examples to show that this result is best possible; (2) if and Δ(G)≥9, then , and we give an infinite family of examples to show that the bound on cannot be increased in general; (3) if G is planar and has girth at least 5, then .     

A Representation of Projection Lattices and Their States in Euclidean Space

We propose a representationr : ∪ Ω → ^ν, where is the collection of closed subspaces of ann-dimensional real, complex, or quaternionic Hilbert space , or equivalently, the projection lattice of this Hilbert space, where Ω is the set of all states ω : → [0, 1]. The value that ω ∈ Ω takes ina ∈ is given by the scalar product of the representative points (r(a) andr(ω)). The representationr(a ∨ b) of the join of two orthogonal elementsa, b ∈ is equal tor(a) + r(b). The convex closure of the representation of Σ, the set of atoms of , is equal to the representation of Ω.     

Boxicity of line graphs

The boxicity of a graph H, denoted by , is the minimum integer k such that H is an intersection graph of axis-parallel k-dimensional boxes in R^k. In this paper we show that for a line graph G of a multigraph, , where Δ(G) denotes the maximum degree of G. Since G is a line graph, Δ(G)≤2(χ(G)−1), where χ(G) denotes the chromatic number of G, and therefore, . For the d-dimensional hypercube Q_d, we prove that . The question of finding a nontrivial lower bound for was left open by Chandran and Sivadasan in [L. Sunil Chandran, Naveen Sivadasan, The cubicity of Hypercube Graphs. Discrete Mathematics 308 (23) (2008) 5795–5800].The above results are consequences of bounds that we obtain for the boxicity of a fully subdivided graph (a graph that can be obtained by subdividing every edge of a graph exactly once).     

More Efficient Algorithm for Ordered Tree Inclusion

Given two ordered trees and , the tree inclusion problem is to determine whether it is possible to obtain from by deleting nodes. Recently, this problem has been recognized as an important primitive in query processing for structured text databases. In this paper we present anO(|leaves()| ||) time andO(|leaves()|min(depth(), |leaves()|)) space algorithm for ordered tree inclusion, by means of a sophisticated bottom-up-matching strategy. Our algorithm improves the previous best one (Kilpeläinen, 1992, Ph.D. thesis, Dept. Computer Science, Univ. Helsinki) that requiresO(|| ||) time andO(||min(depth(), |leaves()|)) space.     

A further refinement of a three critical points theorem

In this paper, we complete the refinement process, made by Ricceri (2009) [4], of a result established by Ricceri (2000) [1], which is one of the most applied abstract multiplicity theorems in the past decade. A sample of application of our new result is as follows.Let (n≥3) be a bounded domain with smooth boundary and let .Then, for each ?>0 small enough, there exists λ_?>0 such that, for every compact interval , there exists ρ>0 with the following property: for every λ∈[a,b] and every continuous function satisfying for some , there exists δ>0 such that, for each ν∈[0,δ], the problem has at least three weak solutions whose norms in are less than ρ.     

On the layered nearest neighbour estimate, the bagged nearest neighbour estimate and the random forest method in regression and classification

Let be identically distributed random vectors in R^d, independently drawn according to some probability density. An observation is said to be a layered nearest neighbour (LNN) of a point if the hyperrectangle defined by and contains no other data points. We first establish consistency results on , the number of LNN of . Then, given a sample of independent identically distributed random vectors from R^d×R, one may estimate the regression function by the LNN estimate , defined as an average over the Y_i’s corresponding to those which are LNN of . Under mild conditions on r, we establish the consistency of towards 0 as n→∞, for almost all and all p≥1, and discuss the links between r_n and the random forest estimates of Breiman (2001) [8]. We finally show the universal consistency of the bagged (bootstrap-aggregated) nearest neighbour method for regression and classification.