Critical groups for complete multipartite graphs and Cartesian products of complete graphs

The critical group of a connected graph is a finite abelian group, whose order is the number of spanning trees in the graph, and which is closely related to the graph Laplacian. Its group structure has been determined for relatively few classes of graphs, e.g., complete graphs and complete bipartite graphs. For complete multipartite graphs , we describe the critical group structure completely. For Cartesian products of complete graphs , we generalize results of H. Bai on the k-dimensional cube, by bounding the number of invariant factors in the critical group, and describing completely its p-primary structure for all primes p that divide none of . © 2003 Wiley Periodicals, Inc. J Graph Theory 44: 231–250, 2003     

The isoperimetric constant of the random graph process

The isoperimetric constant of a graph G on n vertices, i(G), is the minimum of , taken over all nonempty subsets S ⊂ V (G) of size at most n/2, where ∂S denotes the set of edges with precisely one end in S. A random graph process on n vertices, , is a sequence of graphs, where is the edgeless graph on n vertices, and is the result of adding an edge to , uniformly distributed over all the missing edges. The authors show that in almost every graph process equals the minimal degree of as long as the minimal degree is o(log n). Furthermore, it is shown that this result is essentially best possible, by demonstrating that along the period in which the minimum degree is typically Θ(log n), the ratio between the isoperimetric constant and the minimum degree falls from 1 to , its final value. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2008     

A list analogue of equitable coloring

Given lists of available colors assigned to the vertices of a graph G, a list coloring is a proper coloring of G such that the color on each vertex is chosen from its list. If the lists all have size k, then a list coloring is equitable if each color appears on at most vertices. A graph is equitably k-choosable if such a coloring exists whenever the lists all have size k. We prove that G is equitably k-choosable when unless G contains or k is odd and . For forests, the threshold improves to . If G is a 2-degenerate graph (given k ≥ 5) or a connected interval graph (other than ), then G is equitably k-choosable when . © 2003 Wiley Periodicals, Inc. J Graph Theory 44: 166–177, 2003     

Compactness Conditions for the Green Operator in Lp(ℝ) Corresponding to a General Sturm–Liouville Operator

We consider the equation ℝ, where , for ℝ, (ℝ), (ℝ), (ℝ), (ℝ) := C(ℝ)). We give necessary and sufficient conditions under which, regardless of , the following statements hold simultaneously: I) For any (ℝ) Equation (0.1) has a unique solution (ℝ) where $\int ^{\infty}_{-\infty}$ ℝ. II) The operator (ℝ) → (ℝ) is compact. Here is the Green function corresponding to (0.1). This result is applied to study some properties of the spectrum of the Sturm–Liouville operator.     

The cover time of random geometric graphs

We study the cover time of random geometric graphs. Let $I(d)=[0,1]^{d}$ denote the unit torus in d dimensions. Let $D(x,r)$ denote the ball (disc) of radius r. Let $\Upsilon_d$ be the volume of the unit ball $D(0,1)$ in d dimensions. A random geometric graph $G=G(d,r,n)$ in d dimensions is defined as follows: Sample n points V independently and uniformly at random from $I(d)$ . For each point x draw a ball $D(x,r)$ of radius r about x. The vertex set $V(G)=V$ and the edge set $E(G)=\{\{v,w\}: w\ne v,\,w\in D(v,r)\}$ . Let $G(d,r,n),\,d\geq 3$ be a random geometric graph. Let $C_G$ denote the cover time of a simple random walk on G. Let $c>1$ be constant, and let $r=(c\log n/(\Upsilon_dn))^{1/d}$ . Then whp the cover time satisfies © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 38, 324–349, 2011     

Strengths and Weaknesses of LH Arithmetic

In this paper we provide a new arithmetic characterization of the levels of the og‐time hierarchy (LH). We define arithmetic classes and that correspond to ‐LOGTIME and ‐LOGTIME, respectively. We break and into natural hierarchies of subclasses and . We then define bounded arithmetic deduction systems ′ whose ‐definable functions are precisely B( ‐LOGTIME). We show these theories are quite strong in that (1) LIOpen proves for any fixed m that , (2) TAC, a theory that is slightly stronger than ′ whose (LH)‐definable functions are LH, proves LH is not equal to ‐TIME(s) for any m> 0, where 2^s ∈L, s(n) ∈ ω(log n), and (3) TAC proves LH ≠ for all k and m. We then show that the theory TAC cannot prove the collapse of the polynomial hierarchy. Thus any such proof, if it exists, must be argued in a stronger systems than ours.     

Quasilinear degenerate parabolic equations of Kirchhoff type

We investigate the evolution problem where H is a Hilbert space, A is a self‐adjoint linear non‐negative operator on H with domain D(A), and is a continuous function. We prove that if , and , then there exists at least one global solution, which is unique if either m never vanishes, or m is locally Lipschitz continuous. Moreover, we prove that if for all , then this problem is well posed in H. On the contrary, if for some it happens that for all , then this problem has no solution if with β small enough. We apply these results to degenerate parabolic PDEs with non‐local non‐linearities. Copyright © 1999 John Wiley & Sons, Ltd.     

Uniqueness of Integrable Solutions ∇ζ = Gζ, ζ∣Γ = 0 for Integrable Tensor-Coefficients G and Applications to Elasticity

Let be bounded Lipschitz and relatively open. We show that the solution to the linear first order system 1 : (1) vanishes if and , (e.g. ). We prove to be a norm if with , for some p, q > 1 with 1/p + 1/q = 1 and . We give a new proof for the so called ‘in-finitesimal rigid displacement lemma’ in curvilinear coordinates: Let , satisfy for some with . Then there are and a constant skew-symmetric matrix , such that . (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the Points on the Unit Circle with Finite b–Adic Expansions

Let be an arbitrary integer base and let be the number of different prime factors of with , . Further let be the set of points on the unit circle with finite –adic expansions of their coordinates and let be the set of angles of the points . Then is an additive group which is the direct sum of infinite cyclic groups and of the finite cyclic group . If in case of the points of are arranged according to the number of digits of their coordinates, then the arising sequence is uniformly distributed on the unit circle. On the other hand, in case of the only points in are the exceptional points (1, 0), (0, 1), (–1, 0), (0, –1). The proofs are based on a canonical form for all integer solutions of .     

On the Eigenvalues of Half‐Linear Boundary Value Problems

We consider the half‐linear boundary value problem where and the weight function q is assumed to change sign. We prove the existence of two sequences , of eigenvalues and derive asymptotic estimates for as .     

The circular chromatic number of the Mycielskian of G

In a search for triangle-free graphs with arbitrarily large chromatic numbers, Mycielski developed a graph transformation that transforms a graph G into a new graph μ(G), we now call the Mycielskian of G, which has the same clique number as G and whose chromatic number equals χ(G) + 1. Chang, Huang, and Zhu [G. J. Chang, L. Huang, & X. Zhu, Discrete Math, to appear] have investigated circular chromatic numbers of Mycielskians for several classes of graphs. In this article, we study circular chromatic numbers of Mycielskians for another class of graphs G. The main result is that χ_c(μ(G)) = χ(μ(G)), which settles a problem raised in [G. J. Chang, L. Huang, & X. Zhu, Discrete Math, to appear, and X. Zhu, to appear]. As χ_c(G) = and χ(G) = , consequently, there exist graphs G such that χ_c(G) is as close to χ(G) − 1 as you want, but χ_c(μ(G)) = χ(μ(G)). © 1999 John Wiley & Sons, Inc. J Graph Theory 32: 63–71, 1999     

Homomorphism bounds for oriented planar graphs

If is a class of oriented graphs (directed graphs without opposite arcs), then an oriented graph is a homomorphism bound for if there is a homomorphism from each graph in to H. We find some necessary conditions for a graph to be a homomorphism bound for the class of oriented planar graphs and prove that such a graph must have maximum degree at least 16; thus there exists an oriented planar graph with oriented chromatic number at least 17. © 2007 Wiley Periodicals, Inc. J Graph Theory 55: 175–190, 2007     

Comparing Packing Measures to Hausdorff Measures on the Line

For each 0 < s < 1, define where , denote respectively the s‐dimensional packing measure and Hausdorff measure, and the infimum is taken over all the sets E ⊂ R with . In this paper we give a nontrivial estimation of c(s), namely, for each 0 < s < 1, where . As an application, we obtain a lower density theorem for Hausdorff measures.     

Hamilton decompositions of complete multipartite graphs with any 2-factor leave

For m ≥ 1 and p ≥ 2, given a set of integers s₁,…,s_q with for and , necessary and sufficient conditions are found for the existence of a hamilton decomposition of the complete p-partite graph , where U is a 2-factor of consisting of q cycles, the jth cycle having length s_j. This result is then used to completely solve the problem when p = 3, removing the condition that . © 2003 Wiley Periodicals, Inc. J Graph Theory 44: 208–214, 2003     

The size‐Ramsey number of trees

Given a graph G, the size‐Ramsey number $\hat r(G)$ is the minimum number m for which there exists a graph F on m edges such that any two‐coloring of the edges of F admits a monochromatic copy of G. In 1983, J. Beck introduced an invariant β(·) for trees and showed that $\hat r(T) = \Omega (\beta (T))$ . Moreover he conjectured that $\hat r(T) = \Theta (\beta (T))$ . We settle this conjecture by providing a family of graphs and an embedding scheme for trees. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2011     

The color space of a graph

If k is a prime power, and G is a graph with n vertices, then a k‐coloring of G may be considered as a vector in $\mathbb{GF}$(k)ⁿ. We prove that the subspace of $\mathbb{GF}$(3)ⁿ spanned by all 3‐colorings of a planar triangle‐free graph with n vertices has dimension n. In particular, any such graph has at least n − 1 nonequivalent 3‐colorings, and the addition of any edge or any vertex of degree 3 results in a 3‐colorable graph. © 2000 John Wiley & Sons, Inc. J Graph Theory 34: 234–245, 2000     

Optimal Regularity for the No‐Sign Obstacle Problem

In this paper we prove the optimal $C^{1,1}(B_{1/2})$ ‐regularity for a general obstacle‐type problem under the assumption that $f*N$ is $C^{1,1}(B_1)$ , where N is the Newtonian potential. This is the weakest assumption for which one can hope to get $C^{1,1}$ ‐regularity. As a by‐product of the $C^{1,1}$ ‐regularity we are able to prove that, under a standard thickness assumption on the zero set close to a free boundary point $x^0$ , the free boundary is locally a $C^1$ ‐graph close to $x^0$ provided f is Dini. This completely settles the question of the optimal regularity of this problem, which has been the focus of much attention during the last two decades. © 2012 Wiley Periodicals, Inc.     

Coloring edges of embedded graphs

In this paper, we prove that any graph G with maximum degree , which is embeddable in a surface Σ of characteristic χ(Σ) ≤ 1 and satisfies , is class one. © 2000 John Wiley & Sons, Inc. J Graph Theory 35: 197–205, 2000     

Eulerian subgraphs in 3‐edge‐connected graphs and Hamiltonian line graphs

In this paper, we show that if G is a 3‐edge‐connected graph with and , then either G has an Eulerian subgraph H such that , or G can be contracted to the Petersen graph in such a way that the preimage of each vertex of the Petersen graph contains at least one vertex in S. If G is a 3‐edge‐connected planar graph, then for any , G has an Eulerian subgraph H such that . As an application, we obtain a new result on Hamiltonian line graphs. © 2003 Wiley Periodicals, Inc. J Graph Theory 42: 308–319, 2003     

Persistence of activity in threshold contact processes,an “Annealed approximation” of random Boolean networks

We consider a model for gene regulatory networks that is a modification of Kauffmann's J Theor Biol 22 (1969), 437–467 random Boolean networks. There are three parameters: $n = {\rm the}$ number of nodes, $r = {\rm the}$ number of inputs to each node, and $p = {\rm the}$ expected fraction of 1'sin the Boolean functions at each node. Following a standard practice in thephysics literature, we use a threshold contact process on a random graph on n nodes, in which each node has in degree r, to approximate its dynamics. We show that if $r\ge 3$ and $r \cdot 2p(1-p)>1$ , then the threshold contact process persists for a long time, which correspond to chaotic behavior of the Boolean network. Unfortunately, we are only able to prove the persistence time is $\ge \exp(cn^{b(p)})$ with $b(p)>0$ when $r\cdot 2p(1-p)> 1$ , and $b(p)=1$ when $(r-1)\cdot 2p(1-p)>1$ . © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2011