Longest cycles in 3-connected graphs

In this paper, we get the following result: Let G be a 3-connected graph with n vertices. Then , where

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. This is a new lower bound for the circumference c(G) of a 3-connected graph G.     

Directed triangles in directed graphs

We show that each directed graph on n vertices, each with indegree and outdegree at least n/t, where

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, contains a directed circuit of length at most 3.     

Extremal regular graphs for the achromatic number

The problem of constructing (m, n) cages suggests the following class of problems. For a graph parameter θ, determine the minimum or maximum value of p for which there exists a k-regular graph on p points having a given value of θ. The minimization problem is solved here when θ is the achromatic number, denoted by ψ. This result follows from the following main theorem. Let M(p, k) be the maximum value of ψ(G) over all k-regular graphs G with p points, let {x} be the least integer of size at least x, and let be given by ω(k) = {i(ik+1)+1:1i<∞}. Define the function ƒ(p, k) by

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. Then for fixed k2 we have M(p, K=ƒ(p, k) if pω(k) and M(p, k)=ƒ(p,k-1 if pε ω(k) for all p sufficiently large with respect to k.     

Universal H-colorable graphs without a given configuration

For every pair of finite connected graphs F and H, and every positive integer k, we construct a universal graph U with the following properties:

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Particularly, this solves a problem presented in [1] and [2] regarding the chromatic number of a universal graph.     

Atoms of set systems with a fixed number of pairwise unions

For a finite set system with ground set X, we let

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. An atom of H is a nonempty maximal subset C of X such that for all A H, either C A or C ∩ A = 0. We obtain a best possible upper bound for the number of atoms determined by a set system H with H = k and H H = u for all integers k and u. This answers a problem posed by Sós.     

Large 2P3-free graphs with bounded degree

Let ex^* (D;H) be the maximum number of edges in a connected graph with maximum degree D and no induced subgraph H; this is finite if and only if H is a disjoint union of paths. If the largest component of such an H has order m, then ex^*(D; H) = O(D²ex^*(D; P_m)). Constructively, ex^*(D;qP_m) = Θ(gD²ex^*(D;P_m)) if q>1 and m> 2(Θ(gD²) if m = 2). For H = 2P₃ (and D 8), the maximum number of edges is if D is even and

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if D is odd, achieved by a unique extremal graph.     

Strong continuity of the relative rearrangement maps and application to a Galerkin approach for nonlocal problems

We study the strong continuity of the map u   (b_*u, b_*u(| > u(·)|)). Here,

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for σ]0 means Ω[, u_* (respectively, (b|_{u=u*(σ)})*) denotes the decreasing rearrangement of u (respectively b restricted to the set {u = u_*(σ)}) and |E| denotes the Lebesgue measure of a set E included in a domain Ω. The results are useful for solving plasmas physics equations or any nonlocal problems involving the monotone rearrangement, its inverse or its derivatives.     

Choosability of planar graphs

A graph G = G(V, E) with lists L(v), associated with its vertices v V, is called L-list colourable if there is a proper vertex colouring of G in which the colour assigned to a vertex v is chosen from L(v). We say G is k-choosable if there is at least one L-list colouring for every possible list assignment L with L(v) = k v V(G).

Now, let an arbitrary vertex v of G be coloured with an arbitrary colour f of L(v). We investigate whether the colouring of v can be continued to an L-list colouring of the whole graph. G is called free k-choosable if such an L-list colouring exists for every list assignment L (L(v) = k v V(G)), every vertex v and every colour f L(v). We prove the equivalence of the well-known conjecture of Erd s et al. (1979): “Every planar graph is 5-choosable” with the following conjecture: “Every planar graph is free 5-choosable”.     

The chromatic difference sequence of the Cartesian product of graphs

The chromatic difference sequence cds(G) of a graph G with chromatic number n is defined by cds(G) = (a(1), a(2),…, a(n)) if the sum of a(1), a(2),…, a(t) is the maximum number of vertices in an induced t-colorable subgraph of G for t = 1, 2,…, n. The Cartesian product of two graphs G and H, denoted by G□H, has the vertex set V(G□H = V(G) x V(H) and its edge set is given by (x₁, y₁)(x₂, y₂) ε E(G□H) if either x₁ = x₂ and y₁ y₂ ε E(H) or y₁ = y₂ and x₁x₂ ε E(G).

We obtained four main results: the cds of the product of bipartite graphs, the cds of the product of graphs with cds being nondrop flat and first-drop flat, the non-increasing theorem for powers of graphs and cds of powers of circulant graphs.     

Subgraph distances in graphs defined by edge transfers

For two edge-induced subgraphs F and H of the same size in a graph G, the subgraph H can be obtained from F by an edge jump if there exist four distinct vertices u, v, w, and x in G such that uv ε E(F), wx ε E(G) - E(F), and H = F - uv + wx. The subgraph F is j-transformed into H if H can be obtained from F by a sequence of edge jumps. Necessary and sufficient conditions are presented for a graph G to have the property that every edge-induced subgraph of a fixed size in G can be j-transformed into every other edge-induced subgraph of that size. The minimum number of edge jumps required to transform one subgraph into another is called the jump distance. This distance is a metric and can be modeled by a graph. The jump graph J(G) of a graph G is defined as that graph whose vertices are the edges of G and where two vertices of J(G) are adjacent if and only if the corresponding edges of G are independent. For a given graph G, we consider the sequence {{J^k(G)}} of iterated jump graphs and classify each graph as having a convergent, divergent, or terminating sequence.     

Another view on martingale central limit theorems

Based on the martingale version of the Skorokhod embedding Heyde and Brown (1970) established a bound on the rate of convergence in the central limit theorem (CLT) for discrete time martingales having finite moments of order 2+2δ with 0<δ1. An extension for all δ>0 was proved in Haeusler (1988). This paper presents a rather quick access based solely on truncation, optional stopping, and prolongation techniques for martingale difference arrays to obtain other upper bounds for sup

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(φbeing the standard normal d.f.) yielding weak sufficient conditions for the asymptotic normality of . It is shown that our approach also yields two types of martingale central limit theorems with random norming.     

Some results on characterizing the edges of connected graphs with a given domination number

A dominating set for a graph G = (V, E) is a subset of vertices V′ V such that for all v ε V − V′ there exists some u ε V′ for which {v, u} ε E. The domination number of G is the size of its smallest dominating set(s). For a given graph G with minimum size dominating set D, let m₁ (G, D) denote the number of edges that have neither endpoint in D, and let m₂ (G, D) denote the number of edges that have at least one endpoint in D. We characterize the possible values that the pair (m₁ (G, D), m₂ (G, D)) can attain for connected graphs having a given domination number.     

Comparison of two norms of matrices

Any complex n × n matrix A satisfies the inequality

A ₁ ≤ n 1/2 A _d

where .₁ is the trace norm and ._d is the norm defined by

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,

where B is the set of orthonormal bases in the space of n × 1 matrices. The present work is devoted to the study of matrices A satisfying the identity:

A₁ = n1/2 A _d

This paper is a first step towards a characterization of matrices satisfying this identity. Actually, a workable characterization of matrices subject to this condition is obtained only for n = 2. For n = 3, a partial result on nilpotent matrices is presented. Like our previous study (J. Dazord, Linear Algebra Appl. 254 (1997) 67), this study is a continuation of the work of M. Marcus and M. Sandy (M. Marcus and M. Sandy, Linear and Multilinear Algebra 29 (1991) 283). Also this study is related to the work of R. Gabriel on classification of matrices with respect to unitary similarity (see R. Gabriel, J. Riene Angew, Math. 307/308 (1979) 31; R. Gabriel, Math. Z. 200 (1989) 591).     

First inverse moment of a generalized quadratic form

In this paper an approximate expression for the first inverse moment of

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where _k is a Gaussian stationary vector process is derived. This generalized quadratic form is the estimate of the information matrix when using the Recursive Least Squares (RLS) algorithm with forgetting factor. This estimator is commonly used when estimating parameters in time-varying linear stochastic systems.     

Algorithmic complexity of list colorings

Given a graph G = (V,E) and a finite set L(v) at each vertex v ε V, the List Coloring problem asks whether there exists a function f:V → _vεVL(V) such that (i) f(v)εL(v) for each vεV and (ii) f(u) ≠f(v) whenever u, vεV and uvεE. One of our results states that this decision problem remains NP-complete even if all of the followingconditions are met: (1) each set L(v) has at most three elements, (2) each “color” xε_vεVL(v) occurs in at most three sets L(v), (3) each vertex vεV has degree at most three, and (4) G is a planar graph. On the other hand, strengthening any of the assumptions (1)–(3) yields a polynomially solvable problem. The connection between List Coloring and Boolean Satisfiability is discussed, too.     

Bandwidth of the composition of two graphs

The bandwidth B(G) of a graph G is the minimum of the quantity max{|f(x)−f(y)| : xyE(G)} taken over all proper numberings f of G. The composition of two graphs G and H, written as G[H], is the graph with vertex set V(G)×V(H) and with (u₁,v₁) is adjacent to (u₂,v₂) if either u₁ is adjacent to u₂ in G or u₁=u₂ and v₁ is adjacent to v₂ in H. In this paper, we investigate the bandwidth of the composition of two graphs. Let G be a connected graph. We denote the diameter of G by D(G). For two distinct vertices x,yV(G), we define w_G(x,y) as the maximum number of internally vertex-disjoint (x,y)-paths whose lengths are the distance between x and y. We define w(G) as the minimum of w_G(x,y) over all pairs of vertices x,y of G with the distance between x and y is equal to D(G). Let G be a non-complete connected graph and let H be any graph. Among other results, we prove that if |V(G)|=B(G)D(G)−w(G)+2, then B(G[H])=(B(G)+1)|V(H)|−1. Moreover, we show that this result determines the bandwidth of the composition of some classes of graphs composed with any graph.     

Large eigenvalues of the laplacian

Let G be a simple graph on n vertices and let L=L(G) be the Laplacian matrix of G corresponding to some ordering of the vertices. It is known that λ≤n for any eigenvalue λ of L. In this note we characterize when n is an eigenvalue of L with multiplicity m.     

On the planarity of jump graphs

For a graph G of size m1 and edge-induced subgraphs F and H of size k (1km), the subgraph H is said to be obtained from F by an edge jump if there exist four distinct vertices u,v,w, and x in G such that uvE(F), wxE(G)−E(F), and H=F−uv+wx. The minimum number of edge jumps required to transform F into H is the k-jump distance from F to H. For a graph G of size m1 and an integer k with 1km, the k-jump graph J_k(G) is that graph whose vertices correspond to the edge-induced subgraphs of size k of G and where two vertices of J_k(G) are adjacent if and only if the k-jump distance between the corresponding subgraphs is 1. All connected graphs G for which J₂(G) is planar are determined.     

On the minor-minimal 2-connected graphs having a fixed minor

Let H be a graph with κ₁ components and κ₂ blocks, and let G be a minor-minimal 2-connected graph having H as a minor. This paper proves that |E(G)|−|E(H)|(κ₁−1)+β(κ₂−1) for all (,β) such that +β5,2+5β20, and β3. Moreover, if one of the last three inequalities fails, then there are graphs G and H for which the first inequality fails.     

Some results on integral sum graphs

Let Z denote the set of all integers. The integral sum graph of a finite subset S of Z is the graph (S,E) with vertex set S and edge set E such that for u,vS, uvE if and only if u+vS. A graph G is called an integral sum graph if it is isomorphic to the integral sum graph of some finite subset S of Z. The integral sum number of a given graph G, denoted by ζ(G), is the smallest number of isolated vertices which when added to G result in an integral sum graph. Let x denote the least integer not less than the real x. In this paper, we (i) determine the value of ζ(K_n−E(K_r)) for r2n/3−1, (ii) obtain a lower bound for ζ(K_n−E(K_r)) when 2r<2n/3−1 and n5, showing by construction that the bound is sharp when r=2, and (iii) determine the value of ζ(K_r,r) for r2. These results provide partial solutions to two problems posed by Harary (Discrete Math. 124 (1994) 101–108). Finally, we furnish a counterexample to a result on the sum number of K_r,s given by Hartsfiedl and Smyth (Graphs and Matrices, R. Rees (Ed.), Marcel, Dekker, New York, 1992, pp. 205–211).