The Connected Monophonic Number of a Graph

For a connected graph G = (V, E) of order at least two, a chord of a path P is an edge joining two non-adjacent vertices of P. A path P is called a monophonic path if it is a chordless path. A set S of vertices of G is a monophonic set of G if each vertex v of G lies on an x ? y monophonic path for some elements x and y in S. The minimum cardinality of a monophonic set of G is defined as the monophonic number of G, denoted by m(G). A connected monophonic set of G is a monophonic set S such that the subgraph G[S] induced by S is connected. The minimum cardinality of a connected monophonic set of G is the connected monophonic number of G and is denoted by m _c (G). We determine bounds for it and characterize graphs which realize these bounds. For any two vertices u and v in G, the monophonic distance d _m (u, v) from u to v is defined as the length of a longest u ? v monophonic path in G. The monophonic eccentricity e _m (v) of a vertex v in G is the maximum monophonic distance from v to a vertex of G. The monophonic radius rad _m G of G is the minimum monophonic eccentricity among the vertices of G, while the monophonic diameter diam _m G of G is the maximum monophonic eccentricity among the vertices of G. It is shown that for positive integers r, d and n ≥ 5 with r <  d, there exists a connected graph G with rad _m G =  r, diam _m G =  d and m _c (G) =  n. Also, if a,b and p are positive integers such that 2 ≤  a <  b ≤  p, then there exists a connected graph G of order p, m(G) =  a and m _c (G) =  b.     

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.     

On the vanishing of Selmer groups for elliptic curves over ring class fields

Let E/Q be an elliptic curve of conductor N without complex multiplication and let K be an imaginary quadratic field of discriminant D prime to N. Assume that the number of primes dividing N and inert in K is odd, and let Hc be the ring class field of K of conductor c prime to ND with Galois group Gc over K. Fix a complex character χ of Gc. Our main result is that if LK(E,χ,1)≠0 then Selp(E/Hc)χ_⊗W=0 for all but finitely many primes p, where Selp(E/Hc) is the p-Selmer group of E over Hc and W is a suitable finite extension of Zp containing the values of χ. Our work extends results of Bertolini and Darmon to almost all non-ordinary primes p and also offers alternative proofs of a χ-twisted version of the Birch and Swinnerton-Dyer conjecture for E over Hc (Bertolini and Darmon) and of the vanishing of Selp(E/K) for almost all p (Kolyvagin) in the case of analytic rank zero.     

Coincidences of projections and linear n-valued maps of tori

We prove that the Nielsen fixed point number N(φ) of an n-valued map φ:X?X of a compact connected triangulated orientable q-manifold without boundary is equal to the Nielsen coincidence number of the projections of the graph of φ, a subset of X×X, to the two factors. For certain q×q integer matrices A, there exist “linear” n-valued maps Φn,A,σ:Tq?Tq of q-tori that generalize the single-valued maps fA:Tq→Tq induced by the linear transformations TA:Rq→Rq defined by TA(v)=Av. By calculating the Nielsen coincidence number of the projections of its graph, we calculate N(Φn,A,σ) for a large class of linear n-valued maps.     

Covering shadows with a smaller volume

For n?2 a construction is given for convex bodies K and L in Rn such that the orthogonal projection Lu onto the subspace u^⊥ contains a translate of Ku for every direction u, while the volumes of K and L satisfy Vn(K)>Vn(L).A more general construction is then given for n-dimensional convex bodies K and L such that each orthogonal projection Lξ onto a k-dimensional subspace ξ contains a translate of Kξ, while the mth intrinsic volumes of K and L satisfy Vm(K)>Vm(L) for all m>k.For each k=1,…,n, we then define the collection Cn,k to be the closure (under the Hausdorff topology) of all Blaschke combinations of suitably defined cylinder sets (prisms).It is subsequently shown that, if L∈Cn,k, and if the orthogonal projection Lξ contains a translate of Kξ for every k-dimensional subspace ξ of Rn, then Vn(K)?Vn(L).The families Cn,k, called k-cylinder bodies of Rn, form a strictly increasing chain

Cn,1⊂Cn,2⊂?⊂Cn,n−1⊂Cn,n,     

On the Euler genus of a 2-connected graph

The Euler genus of the surface Σ obtained from the sphere by the addition of k crosscaps and h handles is ε(Σ) = k + 2h. For a graph G, the Euler genus ε(G) of G is the smallest Euler genus among all surfaces in which G embeds. The following additivity theorem is proved.Theorem. Suppose G = H ∪ K, where H and K have exactly the vertices v and win common. Then ε(G) = min(ε(H + vw) + ε(K + vw), ε(H) + ε(K) + 2).     

On the computation of the hull number of a graph

Let G be a graph. If u,v∈V(G), a u-vshortest path of G is a path linking u and v with minimum number of edges. The closed interval I[u,v] consists of all vertices lying in some u-v shortest path of G. For S⊆V(G), the set I[S] is the union of all sets I[u,v] for u,v∈S. We say that S is a convex set if I[S]=S. The convex hull of S, denoted I_h[S], is the smallest convex set containing S. A set S is a hull set of G if I_h[S]=V(G). The cardinality of a minimum hull set of G is the hull number of G, denoted by hn(G). In this work we prove that deciding whether hn(G)≤k is NP-complete.We also present polynomial-time algorithms for computing hn(G) when G is a unit interval graph, a cograph or a split graph.     

Bijective linear maps on semimodules spanned by Boolean matrices of fixed rank

Let M_m,n(B) be the semimodule of all m×n Boolean matrices where B is the Boolean algebra with two elements. Let k be a positive integer such that 2?k?min(m,n). Let B(m,n,k) denote the subsemimodule of M_m,n(B) spanned by the set of all rank k matrices. We show that if T is a bijective linear mapping on B(m,n,k), then there exist permutation matrices P and Q such that T(A)=PAQ for all A∈B(m,n,k) or m=n and T(A)=PA^tQ for all A∈B(m,n,k). This result follows from a more general theorem we prove concerning the structure of linear mappings on B(m,n,k) that preserve both the weight of each matrix and rank one matrices of weight k². Here the weight of a Boolean matrix is the number of its nonzero entries.     

Diameter-girth sufficient conditions for optimal extraconnectivity in graphs

For a connected graph G, the rth extraconnectivity κ_r(G) is defined as the minimum cardinality of a cutset X such that all remaining components after the deletion of the vertices of X have at least r+1 vertices. The standard connectivity and superconnectivity correspond to κ₀(G) and κ₁(G), respectively. The minimum r-tree degree of G, denoted by ξ_r(G), is the minimum cardinality of N(T) taken over all trees T⊆G of order |V(T)|=r+1, N(T) being the set of vertices not in T that are neighbors of some vertex of T. When r=1, any such considered tree is just an edge of G. Then, ξ₁(G) is equal to the so-called minimum edge-degree of G, defined as ξ(G)=min{d(u)+d(v)-2:uv∈E(G)}, where d(u) stands for the degree of vertex u. A graph G is said to be optimally r-extraconnected, for short κ_r-optimal, if κ_r(G)?ξ_r(G). In this paper, we present some sufficient conditions that guarantee κ_r(G)?ξ_r(G) for r?2. These results improve some previous related ones, and can be seen as a complement of some others which were obtained by the authors for r=1.     

Steiner intervals and Steiner geodetic numbers in distance-hereditary graphs

A Steiner tree for a set S of vertices in a connected graph G is a connected subgraph of G with a smallest number of edges that contains S. The Steiner interval I(S) of S is the union of all the vertices of G that belong to some Steiner tree for S. If S={u,v}, then I(S)=I[u,v] is called the interval between u and v and consists of all vertices that lie on some shortest u-v path in G. The smallest cardinality of a set S of vertices such that ?_u,v∈SI[u,v]=V(G) is called the geodetic number and is denoted by g(G). The smallest cardinality of a set S of vertices of G such that I(S)=V(G) is called the Steiner geodetic number of G and is denoted by sg(G). We show that for distance-hereditary graphs g(G)?sg(G) but that g(G)/sg(G) can be arbitrarily large if G is not distance hereditary. An efficient algorithm for finding the Steiner interval for a set of vertices in a distance-hereditary graph is described and it is shown how contour vertices can be used in developing an efficient algorithm for finding the Steiner geodetic number of a distance-hereditary graph.     

Extremal sizes of subspace partitions

A subspace partition Π of V?= V(n, q) is a collection of subspaces of V such that each 1-dimensional subspace of V is in exactly one subspace of Π. The size of Π is the number of its subspaces. Let σ _q (n, t) denote the minimum size of a subspace partition of V in which the largest subspace has dimension t, and let ρ _q (n, t) denote the maximum size of a subspace partition of V in which the smallest subspace has dimension t. In this article, we determine the values of σ _q (n, t) and ρ _q (n, t) for all positive integers n and t. Furthermore, we prove that if n ≥?2t, then the minimum size of a maximal partial t-spread in V(n +?t ?1, q) is σ _q (n, t).     

Higher rank numerical ranges and low rank perturbations of quantum channels

For a positive integer k, the rank-k numerical range Λk(A) of an operator A acting on a Hilbert space H of dimension at least k is the set of scalars λ such that PAP=λP for some rank k orthogonal projection P. In this paper, a close connection between low rank perturbation of an operator A and Λk(A) is established. In particular, for 1?r<k it is shown that Λk(A)⊆Λk−r(A+F) for any operator F with rank(F)?r. In quantum computing, this result implies that a quantum channel with a k-dimensional error correcting code under a perturbation of rank at most r will still have a (k−r)-dimensional error correcting code. Moreover, it is shown that if A is normal or if the dimension of A is finite, then Λk(A) can be obtained as the intersection of Λk−r(A+F) for a collection of rank r operators F. Examples are given to show that the result fails if A is a general operator. The closure and the interior of the convex set Λk(A) are completely determined. Analogous results are obtained for Λ_∞(A) defined as the set of scalars λ such that PAP=λP for an infinite rank orthogonal projection P. It is shown that Λ_∞(A) is the intersection of all Λk(A) for k=1,2,…. If A−μI is not compact for all μ∈C, then the closure and the interior of Λ_∞(A) coincide with those of the essential numerical range of A. The situation for the special case when A−μI is compact for some μ∈C is also studied.     

On the chromaticity of complete multipartite graphs with certain edges added

Let P(G,λ) be the chromatic polynomial of a graph G. A graph G is chromatically unique if for any graph H, P(H,λ)=P(G,λ) implies H is isomorphic to G. For integers k≥0, t≥2, denote by K((t−1)×p,p+k) the complete t-partite graph that has t−1 partite sets of size p and one partite set of size p+k. Let K(s,t,p,k) be the set of graphs obtained from K((t−1)×p,p+k) by adding a set S of s edges to the partite set of size p+k such that 〈S〉 is bipartite. If s=1, denote the only graph in K(s,t,p,k) by K⁺((t−1)×p,p+k). In this paper, we shall prove that for k=0,1 and p+k≥s+2, each graph G∈K(s,t,p,k) is chromatically unique if and only if 〈S〉 is a chromatically unique graph that has no cut-vertex. As a direct consequence, the graph K⁺((t−1)×p,p+k) is chromatically unique for k=0,1 and p+k≥3.     

Bipartite rainbow numbers of matchings

Given two graphs G and H, let f(G,H) denote the maximum number c for which there is a way to color the edges of G with c colors such that every subgraph H of G has at least two edges of the same color. Equivalently, any edge-coloring of G with at least rb(G,H)=f(G,H)+1 colors contains a rainbow copy of H, where a rainbow subgraph of an edge-colored graph is such that no two edges of it have the same color. The number rb(G,H) is called the rainbow number ofHwith respect toG, and simply called the bipartite rainbow number ofH if G is the complete bipartite graph K_m,n. Erd?s, Simonovits and Sós showed that rb(K_n,K₃)=n. In 2004, Schiermeyer determined the rainbow numbers rb(K_n,K_k) for all n≥k≥4, and the rainbow numbers rb(K_n,kK₂) for all k≥2 and n≥3k+3. In this paper we will determine the rainbow numbers rb(K_m,n,kK₂) for all k≥1.     

On the low regularity of the Benney-Lin equation

We consider the low regularity of the Benney-Lin equation ut+uux+uxxx+β(uxx+uxxxx)+ηuxxxxx=0. We established the global well posedness for the initial value problem of Benney-Lin equation in the Sobolev spaces Hs(R) for 0?s>−2, improving the well-posedness result of Biagioni and Linares [H.A. Biaginoi, F. Linares, On the Benney-Lin and Kawahara equation, J. Math. Anal. Appl. 211 (1997) 131-152]. For s<−2 we also prove some ill-posedness issues.     

On some friends of the sociable numbers

Let s(n) denote the sum of the proper divisors of n. Set s ₀(n) = n, and for k > 0, put s _k (n) := s(s _k-1(n)) if s _k-1(n) > 0. Thus, perfect numbers are those n with s(n)?=?n and amicable numbers are those n with s(n) ?? n but s ₂(n)?=?n. We prove that for each fixed k ?? 1, the set of n which divide s _k (n) has density zero, and similarly for the set of n for which s _k (n) divides n. These results generalize the theorem of Erd?s that for each fixed k, the set of n for which s _k (n)?=?n has density zero.     

Birational composition of quadratic forms over a local field

Let f(X) and g(Y) be nondegenerate quadratic forms of dimensions m and n, respectively, over K, char K ≠ 2. The problem of birational composition of f(X) and g(Y) is considered: When is the product f(X) · g(Y) birationally equivalent over K to a quadratic form h(Z) over K of dimension m + n? The solution of the birational composition problem for anisotropic quadratic forms over K in the case of m = n = 2 is given. The main result of the paper is the complete solution of the birational composition problem for forms f(X) and g(Y) over a local field P, char P ≠ 2.     

Abelian group matrices over the p-adic and rational integers

Let G be a finite abelian group of order n. Let Z and Q denote the rational integers and rationals, respectively. A group matrix for G over Z (or Q) is an n-square matrix of the form Σ_{g ∈ G}a_gP(g), where a_g ∈ Z (or Q) and P is the regular representation of G so that P(g) is an n-square permutation matrix and P(gh) = P(g)P(h) for all g, h ∈ G. It is known that if M is an arbitrary positive definite unimodular matrix over Z then there exists a matrix A over Q such that M = A^τA, where τ denotes transposition. This paper proves that the exact analogue of this theorem holds if one demands that M and A be group matrices for G over Z and Q, respectively. Furthermore, if M is a group matrix for G over the p-adic integers then necessary and sufficient conditions are given for the existence of a group matrix A for G over the p-adic numbers such that M = A^τA.     

Some permutation problems

Put Z_n = {1, 2,…, n} and let π denote an arbitrary permutation of Z_n. Problem I. Let π = (π(1), π(2), …, π(n)). π has an up, down, or fixed point at a according as a < π(a), a > π(a), or a = π(a). Let $\text{A}\text{(r, s, t)}$ be the number of π ∈ Z_n with r ups, s downs, and t fixed points. Problem II. Consider the triple π^?1(a), a, π(a). Let R denote an up and F a down of π and let B(n, r, s) denote the number of π ∈ Z_n with r occurrences of π^?1(a)RaRπ(a) and s occurrences of π^?1(a)FaFπ(a). Generating functions are obtained for each enumerant as well as for a refinement of the second. In each case use is made of the cycle structure of permutations.