Dimension-2 poset competition numbers and dimension-2 poset double competition numbers

Let D=(V(D),A(D)) be a digraph. The competition graph of D, is the graph with vertex set V(D) and edge set . The double competition graph of D, is the graph with vertex set V(D) and edge set . A poset of dimension at most two is a digraph whose vertices are some points in the Euclidean plane R² and there is an arc going from a vertex (x₁,y₁) to a vertex (x₂,y₂) if and only if x₁>x₂ and y₁>y₂. We show that a graph is the competition graph of a poset of dimension at most two if and only if it is an interval graph, at least half of whose maximal cliques are isolated vertices. This answers an open question on the doubly partial order competition number posed by Cho and Kim. We prove that the double competition graph of a poset of dimension at most two must be a trapezoid graph, generalizing a result of Kim, Kim, and Rho. Some connections are also established between the minimum numbers of isolated vertices required to be added to change a given graph into the competition graph, the double competition graph, of a poset and the minimum sizes of certain intersection representations of that graph.     

Cube intersection concepts in median graphs

In this paper, we study different classes of intersection graphs of maximal hypercubes of median graphs. For a median graph G and k≥0, the intersection graph Q_k(G) is defined as the graph whose vertices are maximal hypercubes (by inclusion) in G, and two vertices H_x and H_y in Q_k(G) are adjacent whenever the intersection H_x∩H_y contains a subgraph isomorphic to Q_k. Characterizations of clique-graphs in terms of these intersection concepts when k>0, are presented. Furthermore, we introduce the so-called maximal 2-intersection graph of maximal hypercubes of a median graph G, denoted , whose vertices are maximal hypercubes of G, and two vertices are adjacent if the intersection of the corresponding hypercubes is not a proper subcube of some intersection of two maximal hypercubes. We show that a graph H is diamond-free if and only if there exists a median graph G such that H is isomorphic to . We also study convergence of median graphs to the one-vertex graph with respect to all these operations.     

Maximal integral point sets in affine planes over finite fields

Motivated by integral point sets in the Euclidean plane, we consider integral point sets in affine planes over finite fields. An integral point set is a set of points in the affine plane over a finite field F_q, where the formally defined squared Euclidean distance of every pair of points is a square in F_q. It turns out that integral point sets over F_q can also be characterized as affine point sets determining certain prescribed directions, which gives a relation to the work of Blokhuis. Furthermore, in one important sub-case, integral point sets can be restated as cliques in Paley graphs of square order.In this article we give new results on the automorphisms of integral point sets and classify maximal integral point sets over F_q for q≤47. Furthermore, we give two series of maximal integral point sets and prove their maximality.     

Further examples of maximal curves

It is shown that the curve over F_q²ⁿ with n≥3 odd, that generalizes Serre’s curve y⁴+y=x³ over F₆₄, is also maximal. We also investigate a family of maximal curves over F_q²ⁿ and provide isomorphisms between these curves.     

Entire choosability of near-outerplane graphs

It is proved that if G is a plane embedding of a K₄-minor-free graph with maximum degree Δ, then G is entirely 7-choosable if Δ≤4 and G is entirely (Δ+2)-choosable if Δ≥5; that is, if every vertex, edge and face of G is given a list of max{7,Δ+2} colours, then every element can be given a colour from its list such that no two adjacent or incident elements are given the same colour. It is proved also that this result holds if G is a plane embedding of a K_2,3-minor-free graph or a -minor-free graph. As a special case this proves that the Entire Coluring Conjecture, that a plane graph is entirely (Δ+4)-colourable, holds if G is a plane embedding of a K₄-minor-free graph, a K_2,3-minor-free graph or a -minor-free graph.     

Classical solutions to differential inclusions with totally disconnected right-hand side

Let F=F(t,x) be a bounded, Hausdorff continuous multifunction with compact, totally disconnected values. Given any y₀∈F(t₀,x₀), we show that the differential inclusion has a globally defined classical solution, with x(t₀)=x₀, .     

Existence-uniqueness results for a class of singular boundary value problems-II

In this paper we establish existence-uniqueness of solution of a class of singular boundary value problem −(p(x)y^′′(x))=q(x)f(x,y) for 0<x?b and y(0)=a, α₁y(b)+β₁y^′(b)=γ₁, where p(x) satisfies (i) p(x)>0 in (0,b), (ii) p(x)∈C¹(0,r), and for some r>b, (iii) is analytic in and q(x) satisfies (i) q(x)>0 in (0,b), (ii) q(x)∈L¹(0,b) and for some r>b, (iii) is analytic in with quite general conditions on f(x,y). Region for multiple solutions have also been determined.     

Approximating reals by sums of two rationals

We generalize Dirichlet's diophantine approximation theorem to approximating any real number α by a sum of two rational numbers with denominators 1?q₁,q₂?N. This turns out to be related to the congruence equation problem with 1?x,y?q1/2+?.     

The treewidth and pathwidth of hypercubes

The d-dimensional hypercube, H_d, is the graph on 2^d vertices, which correspond to the 2^dd-vectors whose components are either 0 or 1, two of the vertices being adjacent when they differ in just one coordinate. The notion of Hamming graphs (denoted by ) generalizes the notion of hypercubes: The vertices correspond to the q^dd-vectors where the components are from the set {0,1,2,…,q-1}, and two of the vertices are adjacent if and only if the corresponding vectors differ in exactly one component. In this paper we show that the and the .     

Eigenvalues and edge-connectivity of regular graphs

In this paper, we show that if the second largest eigenvalue of a d-regular graph is less than , then the graph is k-edge-connected. When k is 2 or 3, we prove stronger results. Let ρ(d) denote the largest root of x³-(d-3)x²-(3d-2)x-2=0. We show that if the second largest eigenvalue of a d-regular graph G is less than ρ(d), then G is 2-edge-connected and we prove that if the second largest eigenvalue of G is less than , then G is 3-edge-connected.     

On the spanning fan-connectivity of graphs

Let G be a graph. The connectivity of G, κ(G), is the maximum integer k such that there exists a k-container between any two different vertices. A k-container of G between u and v, C_k(u,v), is a set of k-internally-disjoint paths between u and v. A spanning container is a container that spans V(G). A graph G is k^∗-connected if there exists a spanning k-container between any two different vertices. The spanning connectivity of G, κ^∗(G), is the maximum integer k such that G is w^∗-connected for 1≤w≤k if G is 1^∗-connected.Let x be a vertex in G and let U={y₁,y₂,…,y_k} be a subset of V(G) where x is not in U. A spanningk−(x,U)-fan, F_k(x,U), is a set of internally-disjoint paths {P₁,P₂,…,P_k} such that P_i is a path connecting x to y_i for 1≤i≤k and . A graph G is k^∗-fan-connected (or -connected) if there exists a spanning F_k(x,U)-fan for every choice of x and U with |U|=k and x∉U. The spanning fan-connectivity of a graph G, , is defined as the largest integer k such that G is -connected for 1≤w≤k if G is -connected.In this paper, some relationship between κ(G), κ^∗(G), and are discussed. Moreover, some sufficient conditions for a graph to be -connected are presented. Furthermore, we introduce the concept of a spanning pipeline-connectivity and discuss some sufficient conditions for a graph to be k^∗-pipeline-connected.     

Perfectness and imperfectness of unit disk graphs on triangular lattice points

Given a finite set of 2-dimensional points P⊆R² and a positive real d, a unit disk graph, denoted by (P,d), is an undirected graph with vertex set P such that two vertices are adjacent if and only if the Euclidean distance between the pair is less than or equal to d. Given a pair of non-negative integers m and n, P(m,n) denotes a subset of 2-dimensional triangular lattice points defined by where . Let T_m,n(d) be a unit disk graph defined on a vertex set P(m,n) and a positive real d. Let be the kth power of T_m,n(1).In this paper, we show necessary and sufficient conditions that [ is perfect] and/or [ is perfect], respectively. These conditions imply polynomial time approximation algorithms for multicoloring (T_m,n(d),w) and .     

3-trees with few vertices of degree 3 in circuit graphs

A circuit graph(G,C) is a 2-connected plane graph G with an outer cycle C such that from each inner vertex v, there are three disjoint paths to C. In this paper, we shall show that a circuit graph with n vertices has a 3-tree (i.e., a spanning tree with maximum degree at most 3) with at most vertices of degree 3. Our estimation for the number of vertices of degree 3 is sharp. Using this result, we prove that a 3-connected graph with n vertices on a surface F_χ with Euler characteristic χ≥0 has a 3-tree with at most vertices of degree 3, where c_χ is a constant depending only on F_χ.     

Expansions in non-integer bases: Lower, middle and top orders

Let q∈(1,2); it is known that each x∈[0,1/(q−1)] has an expansion of the form with an∈{0,1}. It was shown in [P. Erd?s, I. Joó, V. Komornik, Characterization of the unique expansions and related problems, Bull. Soc. Math. France 118 (1990) 377-390] that if , then each x∈(0,1/(q−1)) has a continuum of such expansions; however, if , then there exist infinitely many x having a unique expansion [P. Glendinning, N. Sidorov, Unique representations of real numbers in non-integer bases, Math. Res. Lett. 8 (2001) 535-543]. In the present paper we begin the study of parameters q for which there exists x having a fixed finite number m>1 of expansions in base q. In particular, we show that if q<q₂=1.71…, then each x has either 1 or infinitely many expansions, i.e., there are no such q in . On the other hand, for each m>1 there exists γm>0 such that for any q∈(2−γm,2), there exists x which has exactly m expansions in base q.     

Generalization of transitive fraternal augmentations for directed graphs and its applications

Let be a directed graph. A transitive fraternal augmentation of is a directed graph with the same vertex set, including all the arcs of and such that for any vertices x,y,z,

1.

if and then or (fraternity);

2.

if and then (transitivity).

In this paper, we explore some generalization of the transitive fraternal augmentations for directed graphs and its applications. In particular, we show that the 2-coloring number col₂(G)≤O(∇₁(G)∇₀(G)²), where ∇_k(G) (k≥0) denotes the greatest reduced average density with depth k of a graph G; we give a constructive proof that ∇_k(G) bounds the distance (k+1)-coloring number col_k+1(G) with a function f(∇_k(G)). On the other hand, ∇_k(G)≤(col_2k+1(G))^2k+1. We also show that an inductive generalization of transitive fraternal augmentations can be used to study nonrepetitive colorings of graphs.     

Minimizing a monotone concave function with laminar covering constraints

Let V be a finite set with |V|=n. A family F⊆2^V is called laminar if for all two sets X,Y∈F, X∩Y≠∅ implies X⊆Y or X⊇Y. Given a laminar family F, a demand function , and a monotone concave cost function , we consider the problem of finding a minimum-cost such that x(X)?d(X) for all X∈F. Here we do not assume that the cost function F is differentiable or even continuous. We show that the problem can be solved in O(n²q) time if F can be decomposed into monotone concave functions by the partition of V that is induced by the laminar family F, where q is the time required for the computation of F(x) for any . We also prove that if F is given by an oracle, then it takes Ω(n²q) time to solve the problem, which implies that our O(n²q) time algorithm is optimal in this case. Furthermore, we propose an algorithm if F is the sum of linear cost functions with fixed setup costs. These also make improvements in complexity results for source location and edge-connectivity augmentation problems in undirected networks. Finally, we show that in general our problem requires Ω(2^n/2q) time when F is given implicitly by an oracle, and that it is NP-hard if F is given explicitly in a functional form.     

Liouville type theorems for p-harmonic maps

Let M be a complete Riemannian manifold and let N be a Riemannian manifold of non-positive sectional curvature. Assume that at all x∈M and at some point x₀∈M, where μ₀>0 is the least eigenvalue of the Laplacian acting on L²-functions on M. Let 2?q?p. Then any q-harmonic map of finite q-energy is constant. Moreover, if N is a Riemannian manifold of non-positive scalar curvature, then any q-harmonic morphism of finite q-energy is constant.     

On the diameter and girth of a zero-divisor graph

For a commutative ring R with zero-divisors Z(R), the zero-divisor graph of R is Γ(R)=Z(R)−{0}, with distinct vertices x and y adjacent if and only if xy=0. In this paper, we characterize when either or . We then use these results to investigate the diameter and girth for the zero-divisor graphs of polynomial rings, power series rings, and idealizations.