Kernels and some operations in edge-coloured digraphs

Let D be an edge-coloured digraph, V(D) will denote the set of vertices of D; a set N⊆V(D) is said to be a kernel by monochromatic paths of D if it satisfies the following two conditions: For every pair of different vertices u,v∈N there is no monochromatic directed path between them and; for every vertex x∈V(D)−N there is a vertex y∈N such that there is an xy-monochromatic directed path.In this paper we consider some operations on edge-coloured digraphs, and some sufficient conditions for the existence or uniqueness of kernels by monochromatic paths of edge-coloured digraphs formed by these operations from another edge-coloured digraphs.     

Induced path transit function, monotone and Peano axioms

The induced path transit function J(u,v) in a graph consists of the set of all vertices lying on any induced path between the vertices u and v. A transit function J satisfies monotone axiom if x,y∈J(u,v) implies J(x,y)⊆J(u,v). A transit function J is said to satisfy the Peano axiom if, for any u,v,w∈V,x∈J(v,w), y∈J(u,x), there is a z∈J(u,v) such that y∈J(w,z). These two axioms are equivalent for the induced path transit function of a graph. Planar graphs for which the induced path transit function satisfies the monotone axiom are characterized by forbidden induced subgraphs.     

Triangle path transit functions, betweenness and pseudo-modular graphs

The geodesic and induced path transit functions are the two well-studied interval functions in graphs. Two important transit functions related to the geodesic and induced path functions are the triangle path transit functions which consist of all vertices on all u,v-shortest (induced) paths or all vertices adjacent to two adjacent vertices on all u,v-shortest (induced) paths, for any two vertices u and v in a connected graph G. In this paper we study the two triangle path transit functions, namely the I^Δ and J^Δ on G. We discuss the betweenness axioms, for both triangle path transit functions. Also we present a characterization of pseudo-modular graphs using the transit function I^Δ by forbidden subgraphs.     

Identifying codes and locating-dominating sets on paths and cycles

Let G=(V,E) be a graph and let r≥1 be an integer. For a set D⊆V, define N_r[x]={y∈V:d(x,y)≤r} and D_r(x)=N_r[x]∩D, where d(x,y) denotes the number of edges in any shortest path between x and y. D is known as an r-identifying code (r-locating-dominating set, respectively), if for all vertices x∈V (x∈V?D, respectively), D_r(x) are all nonempty and different. Roberts and Roberts [D.L. Roberts, F.S. Roberts, Locating sensors in paths and cycles: the case of 2-identifying codes, European Journal of Combinatorics 29 (2008) 72-82] provided complete results for the paths and cycles when r=2. In this paper, we provide results for a remaining open case in cycles and complete results in paths for r-identifying codes; we also give complete results for 2-locating-dominating sets in cycles, which completes the results of Bertrand et al. [N. Bertrand, I. Charon, O. Hudry, A. Lobstein, Identifying and locating-dominating codes on chains and cycles, European Journal of Combinatorics 25 (2004) 969-987].     

The profile of the Cartesian product of graphs

Given a graph G, a proper labelingf of G is a one-to-one function from V(G) onto {1,2,…,|V(G)|}. For a proper labeling f of G, the profile widthw_f(v) of a vertex v is the minimum value of f(v)−f(x), where x belongs to the closed neighborhood of v. The profile of a proper labelingfofG, denoted by P_f(G), is the sum of all the w_f(v), where v∈V(G). The profile ofG is the minimum value of P_f(G), where f runs over all proper labeling of G. In this paper, we show that if the vertices of a graph G can be ordered to satisfy a special neighborhood property, then so can the graph G×Q_n. This can be used to determine the profile of Q_n and K_m×Q_n.     

Minimal rankings and the arank number of a path

Given a graph G, a function f:V(G)→{1,2,…,k} is a k-ranking of G if f(u)=f(v) implies every u-v path contains a vertex w such that f(w)>f(u). A k-ranking is minimal if the reduction of any label greater than 1 violates the described ranking property. The arank number of a graph, denoted ψ_r(G), is the largest k such that G has a minimal k-ranking. We present new results involving minimal k-rankings of paths. In particular, we determine ψ_r(P_n), a problem posed by Laskar and Pillone in 2000.     

Nonlinear dissipative wave equations with space-time dependent potential

We study the long time behavior of solutions for damped wave equations with absorption. These equations are generally accepted as models of wave propagation in heterogeneous media with space-time dependent friction a(t,x)u_t and nonlinear absorption |u|^p−1u (Ikawa (2000) [17]). We consider 1<p<(n+2)/(n−2) and separable a(t,x)=λ(x)η(t) with λ(x)∼(1+|x|)^−α and η(t)∼(1+t)^−β satisfying conditions (A1) or (A2) which are given. The main results are precise decay estimates for the energy, L² and L^p+1 norms of solutions. We also observe the following behavior: if α∈[0,1), β∈(−1,1) and 0<α+β<1, there are three different regions for the decay of solutions depending on p; if α∈(−∞,0) and β∈(−1,1), there are only two different regions for the decay of the solutions depending on p.     

The multiple originator broadcasting problem in graphs

Given a graph G and a vertex subset S of V(G), the broadcasting time with respect toS, denoted by b(G,S), is the minimum broadcasting time when using S as the broadcasting set. And the k-broadcasting number, denoted by b_k(G), is defined by b_k(G)=min{b(G,S)|S⊆V(G),|S|=k}.Given a graph G and two vertex subsets S, S^′ of V(G), define , d(S,S^′)=min{d(u,v)|u∈S, v∈S^′}, and for all v∈V(G). For all k, 1?k?|V(G)|, the k-radius of G, denoted by r_k(G), is defined as r_k(G)=min{d(G,S)|S⊆V(G), |S|=k}.In this paper, we study the relation between the k-radius and the k-broadcasting numbers of graphs. We also give the 2-radius and the 2-broadcasting numbers of the grid graphs, and the k-broadcasting numbers of the complete n-partite graphs and the hypercubes.     

Spanning trails containing given edges

A graph G is Eulerian-connected if for any u and v in V(G), G has a spanning (u,v)-trail. A graph G is edge-Eulerian-connected if for any e^′ and e^″ in E(G), G has a spanning (e^′,e^″)-trail. For an integer r?0, a graph is called r-Eulerian-connected if for any X⊆E(G) with |X|?r, and for any , G has a spanning (u,v)-trail T such that X⊆E(T). The r-edge-Eulerian-connectivity of a graph can be defined similarly. Let θ(r) be the minimum value of k such that every k-edge-connected graph is r-Eulerian-connected. Catlin proved that θ(0)=4. We shall show that θ(r)=4 for 0?r?2, and θ(r)=r+1 for r?3. Results on r-edge-Eulerian connectivity are also discussed.     

Bounds on isoperimetric values of trees

Let G=(V,E) be a finite, simple and undirected graph. For S⊆V, let δ(S,G)={(u,v)∈E:u∈S and v∈V−S} be the edge boundary of S. Given an integer i, 1≤i≤|V|, let the edge isoperimetric value of G at i be defined as b_e(i,G)=min_S⊆V;|S|=i|δ(S,G)|. The edge isoperimetric peak of G is defined as b_e(G)=max_1≤j≤|V|b_e(j,G). Let b_v(G) denote the vertex isoperimetric peak defined in a corresponding way. The problem of determining a lower bound for the vertex isoperimetric peak in complete t-ary trees was recently considered in [Y. Otachi, K. Yamazaki, A lower bound for the vertex boundary-width of complete k-ary trees, Discrete Mathematics, in press (doi:10.1016/j.disc.2007.05.014)]. In this paper we provide bounds which improve those in the above cited paper. Our results can be generalized to arbitrary (rooted) trees.The depth d of a tree is the number of nodes on the longest path starting from the root and ending at a leaf. In this paper we show that for a complete binary tree of depth d (denoted as ), and where c₁, c₂ are constants. For a complete t-ary tree of depth d (denoted as ) and d≥clogt where c is a constant, we show that and where c₁, c₂ are constants. At the heart of our proof we have the following theorem which works for an arbitrary rooted tree and not just for a complete t-ary tree. Let T=(V,E,r) be a finite, connected and rooted tree — the root being the vertex r. Define a weight function w:V→N where the weight w(u) of a vertex u is the number of its successors (including itself) and let the weight index η(T) be defined as the number of distinct weights in the tree, i.e η(T)=|{w(u):u∈V}|. For a positive integer k, let ?(k)=|{i∈N:1≤i≤|V|,b_e(i,G)≤k}|. We show that .     

On island sequences of labelings with a condition at distance two

An L(2,1)-labeling of a graph G is a function f from the vertex set of G to the set of nonnegative integers such that |f(x)−f(y)|≥2 if d(x,y)=1, and |f(x)−f(y)|≥1 if d(x,y)=2, where d(x,y) denotes the distance between the pair of vertices x,y. The lambda number of G, denoted λ(G), is the minimum range of labels used over all L(2,1)-labelings of G. An L(2,1)-labeling of G which achieves the range λ(G) is referred to as a λ-labeling. A hole of an L(2,1)-labeling is an unused integer within the range of integers used. The hole index of G, denoted ρ(G), is the minimum number of holes taken over all its λ-labelings. An island of a given λ-labeling of G with ρ(G) holes is a maximal set of consecutive integers used by the labeling. Georges and Mauro [J.P. Georges, D.W. Mauro, On the structure of graphs with non-surjective L(2,1)-labelings, SIAM J. Discrete Math. 19 (2005) 208-223] inquired about the existence of a connected graph G with ρ(G)≥1 possessing two λ-labelings with different ordered sequences of island cardinalities. This paper provides an infinite family of such graphs together with their lambda numbers and hole indices. Key to our discussion is the determination of the path covering number of certain 2-sparse graphs, that is, graphs containing no pair of adjacent vertices of degree greater than 2.     

Existence and porosity for a class of perturbed optimization problems in Banach spaces

Let X be a Banach space and Z a nonempty closed subset of X. Let be an upper semicontinuous function bounded from above. This paper is concerned with the perturbed optimization problem supz∈Z{J(z)+‖x−z‖}, which is denoted by (x,J)-sup. We shall prove in the present paper that if Z is a closed boundedly relatively weakly compact nonempty subset, then the set of all x∈X for which the problem (x,J)-sup has a solution is a dense Gδ-subset of X. In the case when X is uniformly convex and J is bounded, we will show that the set of all points x in X for which there does not exist z₀∈Z such that J(z₀)+‖x−z₀‖=supz∈Z{J(z)+‖x−z‖} is a σ-porous subset of X and the set of all points x∈X?Z⁰ such that there exists a maximizing sequence of the problem (x,J)-sup which has no convergent subsequence is a σ-porous subset of X?Z⁰, where Z⁰ denotes the set of all z∈Z such that z is in the solution set of (z,J)-sup.     

Well-posedness of a class of perturbed optimization problems in Banach spaces

Let X be a Banach space and Z a nonempty subset of X. Let J:Z→R be a lower semicontinuous function bounded from below and p?1. This paper is concerned with the perturbed optimization problem of finding z₀∈Z such that ‖x−z₀p^‖+J(z₀)=infz∈Z{‖x−zp^‖+J(z)}, which is denoted by minJ(x,Z). The notions of the J-strictly convex with respect to Z and of the Kadec with respect to Z are introduced and used in the present paper. It is proved that if X is a Kadec Banach space with respect to Z and Z is a closed relatively boundedly weakly compact subset, then the set of all x∈X for which every minimizing sequence of the problem minJ(x,Z) has a converging subsequence is a dense Gδ-subset of X?Z₀, where Z₀ is the set of all points z∈Z such that z is a solution of the problem minJ(z,Z). If additionally p>1 and X is J-strictly convex with respect to Z, then the set of all x∈X for which the problem minJ(x,Z) is well-posed is a dense Gδ-subset of X?Z₀.     

A distance-labelling problem for hypercubes

Let i₁≥i₂≥i₃≥1 be integers. An L(i₁,i₂,i₃)-labelling of a graph G=(V,E) is a mapping ?:V→{0,1,2,…} such that |?(u)−?(v)|≥i_t for any u,v∈V with d(u,v)=t, t=1,2,3, where d(u,v) is the distance in G between u and v. The integer ?(v) is called the label assigned to v under ?, and the difference between the largest and the smallest labels is called the span of ?. The problem of finding the minimum span, λ_i1,i2,i3(G), over all L(i₁,i₂,i₃)-labellings of G arose from channel assignment in cellular communication systems, and the related problem of finding the minimum number of labels used in an L(i₁,i₂,i₃)-labelling was originated from recent studies on the scalability of optical networks. In this paper we study the L(i₁,i₂,i₃)-labelling problem for hypercubes Q_d (d≥3) and obtain upper and lower bounds on λ_i1,i2,i3(Q_d) for any (i₁,i₂,i₃).     

Homothetic interval orders

We give a characterization of the non-empty binary relations ? on a N^*-set A such that there exist two morphisms of N^*-sets u₁,u₂:A→R₊ verifying u₁?u₂ and x?y⇔u₁(x)>u₂(y). They are called homothetic interval orders. If ? is a homothetic interval order, we also give a representation of ? in terms of one morphism of N^*-sets u:A→R₊ and a map such that x?y⇔σ(x,y)u(x)>u(y). The pairs (u₁,u₂) and (u,σ) are “uniquely” determined by ?, which allows us to recover one from each other. We prove that ? is a semiorder (resp. a weak order) if and only if σ is a constant map (resp. σ=1). If moreover A is endowed with a structure of commutative semigroup, we give a characterization of the homothetic interval orders ? represented by a pair (u,σ) so that u is a morphism of semigroups.     

An extremal problem on non-full colorable graphs

For a given graph G of order n, a k-L(2,1)-labelling is defined as a function f:V(G)→{0,1,2,…k} such that |f(u)-f(v)|?2 when d_G(u,v)=1 and |f(u)-f(v)|?1 when d_G(u,v)=2. The L(2,1)-labelling number of G, denoted by λ(G), is the smallest number k such that G has a k-L(2,1)-labelling. The hole index ρ(G) of G is the minimum number of integers not used in a λ(G)-L(2,1)-labelling of G. We say G is full-colorable if ρ(G)=0; otherwise, it will be called non-full colorable. In this paper, we consider the graphs with λ(G)=2m and ρ(G)=m, where m is a positive integer. Our main work generalized a result by Fishburn and Roberts [No-hole L(2,1)-colorings, Discrete Appl. Math. 130 (2003) 513-519].     

Spectral conditions for admissibility of evolution equations in Hilbert space

We study properties of solutions of the evolution equation , where B is a closable operator on the space AP(R,H) of almost periodic functions with values in a Hilbert space H such that B commutes with translations. The operator B generates a family of closed operators on H such that (whenever eiλtx∈D(B)). For a closed subset Λ⊂R, we prove that the following properties (i) and (ii) are equivalent: (i) for every function f∈AP(R,H) such that σ(f)⊆Λ, there exists a unique mild solution u∈AP(R,H) of Eq. (∗) such that σ(u)⊆Λ; (ii) is invertible for all λ∈Λ and .     

More on “Connected (n, m)-graphs with minimum and maximum zeroth-order general Randi? index”

Let G be a graph and d(u) denote the degree of a vertex u in G. The zeroth-order general Randi? index ⁰R_α(G) of the graph G is defined as ∑_u∈V(G)d(u)^α, where the summation goes over all vertices of G and α is an arbitrary real number. In this paper we correct the proof of the main Theorem 3.5 of the paper by Hu et al. [Y. Hu, X. Li, Y. Shi, T. Xu, Connected (n,m)-graphs with minimum and maximum zeroth-order general Randi? index, Discrete Appl. Math. 155 (8) (2007) 1044-1054] and give a more general Theorem. We finally characterize ¹ for α<0 the connected G(n,m)-graphs with maximum value ⁰R_α(G(n,m)), where G(n,m) is a simple connected graph with n vertices and m edges.     

On disconnected cuts and separators

For a connected graph G=(V,E), a subset U⊆V is called a disconnected cut if U disconnects the graph, and the subgraph induced by U is disconnected as well. A natural condition is to impose that for any u∈U, the subgraph induced by (V?U)∪{u} is connected. In that case, U is called a minimal disconnected cut. We show that the problem of testing whether a graph has a minimal disconnected cut is NP-complete. We also show that the problem of testing whether a graph has a disconnected cut separating two specified vertices, s and t, is NP-complete.     

Mapping properties for oscillatory integrals in d-dimensions

For aj,bj?1, j=1,2,…,d, we prove that the operator maps into itself for , where , and k(x,y)=φ(x,y)eig(x,y), φ(x,y) satisfies (1.2) (e.g. φ(x,y)=|x−y|^iτ,τ real) and the phase g(x,y)=xa⋅yb. We study operators with more general phases and for these operators we require that aj,bj>1, j=1,2,…,d, or al=bl?1 for some l∈{1,2,…,d}.