Ore-type conditions for bipartite graphs containing hexagons

Let G=(V₁,V₂;E) be a bipartite graph with |V₁|=|V₂|=3k, where k>0. In this paper it is proved that if d(x)+d(y)≥4k−1 for every pair of nonadjacent vertices x∈V₁, y∈V₂, then G contains k−1 independent cycles of order 6 and a path of order 6 such that all of them are independent. Furthermore, if d(x)+d(y)≥4k for every pair of nonadjacent vertices x∈V₁, y∈V₂ and k>2, then G contains k−2 independent cycles of order 6 and a cycle of order 12 such that all of them are independent.     

Implicitly defined optimization problems

This paper considers the solution of the problem: inff[y, x(y)] s.t.y [y, x(y)] E ^k , wherex(y) solves: minF(x, y) s.t.x R(x, y) E ⁿ . In order to obtain local solutions, a first-order algorithm, which uses {dx(y)/dy} for solving a special case of the implicitly definedy-problem, is given. The derivative is obtained from {dx(y, r)/dy}, wherer is a penalty function parameter and {x(y, r)} are approximations to the solution of thex-problem given by a sequential minimization algorithm. Conditions are stated under whichx(y, r) and {dx(y, r)/dy} exist. The computation of {dx(y, r)/dy} requires the availability of _y F(x, y) and the partial derivatives of the other functions defining the setR(x, y) with respect to the parametersy.Research sponsored by National Science Foundation Grant ECS-8709795 and Office of Naval Research Contract N00014-89-J-1537. We thank the referees for constructive comments on an earlier version of this paper.     

Graphs whose every transitive orientation contains almost every relation

Given a graphG onn vertices and a total ordering ≺ ofV(G), the transitive orientation ofG associated with ≺, denotedP(G; ≺), is the partial order onV(G) defined by settingx<y inP(G; ≺) if there is a pathx=x ₁ x ₂…x _r=y inG such thatx ₁ ≺x _j for 1≦i<j≦r. We investigate graphsG such that every transitive orientation ofG contains ₂ ⁿ −o(n ²) relations. We prove that almost everyG _n,p satisfies this requirement if , but almost noG _n,p satisfies the condition if (pn log log logn)/(logn log logn) is bounded. We also show that every graphG withn vertices and at mostcn logn edges has some transitive orientation with fewer than ₂ ⁿ −δ(c)n ² relations. Partially supported by MCS Grant 8104854.     

On the average distance property and certain energy integrals

One of our main results is the following: LetX be a compact connected subset of the Euclidean spaceR ⁿ andr(X, d ₂) the rendezvous number ofX, whered ₂ denotes the Euclidean distance inR ⁿ . (The rendezvous numberr(X, d ₂) is the unique positive real number with the property that for each positive integern and for all (not necessarily distinct)x ₁,x ₂,...,x _n inX, there exists somex inX such that .) Then there exists some regular Borel probability measure μ₀ onX such that the value of ∫ _X d ₂(x, y)dμ₀ (y) is independent of the choicex inX, if and only ifr(X, d ₂) = sup_μ ∫ _X ∫ _X d ₂(x, y)dμ(x)dμ(y), where the supremum is taken over all regular Borel probability measures μ onX.     

The reverse spelling of an FPrt-universal word in two letters

ForX a set the expression Prt(X) denotes the composition monoid of all functionsf X ×X. Fork a positive integer the letterk denotes also the set of all nonnegative integers less thank. Whenk > 1 the expression r_k denotes the connected injective element {<i, i + 1>i k – 1} in Prt (k). We show for every word w=w(x,y) in a two-letter alphabet that if the equation w(x, y)=r_k has a solution =y) ²Prt(k) then ¯w(x,y)=r_k also has a solution in²Prt(k), where ¯w is the word obtained by spelling the wordw backwards. It is a consequence of this theorem that if for every finite setX and for everyf Prt(X) the equation w(x,y)=f has a solution in²Prt(X) then for every suchX andf the equation ¯w(x, y)=f has a solution in²Prt(X).Presented by J. Mycielski.     

A Degree Sum Condition Concerning the Connectivity and the Independence Number of a Graph

Let G be a graph and S ⊂ V(G). We denote by α(S) the maximum number of pairwise nonadjacent vertices in S. For x, y ∈ V(G), the local connectivity κ(x, y) is defined to be the maximum number of internally-disjoint paths connecting x and y in G. We define . In this paper, we show that if κ(S) ≥ 3 and for every independent set {x ₁, x ₂, x ₃, x ₄} ⊂ S, then G contains a cycle passing through S. This degree condition is sharp and this gives a new degree sum condition for a 3-connected graph to be hamiltonian.     

On the graph of large distances

For a setS of points in the plane, letd ₁>d ₂>... denote the different distances determined byS. Consider the graphG(S, k) whose vertices are the elements ofS, and two are joined by an edge iff their distance is at leastd _k . It is proved that the chromatic number ofG(S, k) is at most 7 if |S|constk ². IfS consists of the vertices of a convex polygon and |S|constk ², then the chromatic number ofG(S, k) is at most 3. Both bounds are best possible. IfS consists of the vertices of a convex polygon thenG(S, k) has a vertex of degree at most 3k – 1. This implies that in this case the chromatic number ofG(S, k) is at most 3k. The best bound here is probably 2k+1, which is tight for the regular (2k+1)-gon.     

A generalization of the Bernoulli model occurring in order statistics. I.

Let X=(x₁, ..., x_n) and Y=(y₁, ..., y_m) be independent samples from populations G_x and G_y, x⁽¹⁾ ,... x⁽ⁿ⁾ be ordered statistics constructed from the sample X. A model of trials associated with the occurrence of dependent events A_k={y_k (x⁽ⁱ⁾}, x^(j), i < j, k=1, 2, ..., m, where x⁽ⁱ⁾, x^(j) are order statistics, is considered. This model is a generalization of the Bernoulli model. Distribution of frequencies of occurrences of events A_k and the limit theorems which describe asymptotic properties of these frequencies are investigated.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 4, pp. 518–528, April, 1990.     

Graphs isomorphic to subgraphs of their iterated line graphs

A graphG is said to be embeddable into a graphH, if there is an isomorphism ofG into a subgraph ofH. It is shown in this paper that every unicycle or tree which is neither a path norK _1,3 embeds in itsn-th iterated line graph forn1 or 2, 3, and that every other connected graph that embeds in itsn-th iterated line graph may be constructed from such an embedded unicycle or tree in a natural way. A special kind of embedding of graph into itsn-th iterated line graph, called incidence embedding, is studied. Moreover, it is shown that for every positive integerk, there exists a graphG such that (G) = , where (G) is the leastn1 for whichG embeds inL ⁿ(G).     

On the dimension to represent a graph by a unit distance graph

It is proved that if a graphG has maximum degreed, then its vertices can be represented by distinct unit vectors inR ^2d so that two vectors are orthogonal if and only if the corresponding vertices are adjacent. As a corollary it follows that if a graph has maximum degreed, then it is isomorphic to a unit distance graph inR ^2d.     

Gitterpunkte unter algebraischen Kurven

Lattice points below Algebraic Curves. A generalization of the classical circle problem is treated. An asymptotic formula for the number of lattice points in a region whose boundary is an algebraic curve is obtained. This gives a mean value formula for the number of representations of the positive integers in the formn=g(x,y), whereg is a polynomial with coefficients 0 and leading terma _d0x^d*a_0dy^d. The caseg(x,y)=p₁(x)+p₂(y) was considered inKuba andNowak [4], andKuba [5]. The discrete Hardy-Littlewood method is used along with Rouché's theorem.

     

Average distance,minimum degree,and spanning trees

The average distance μ(G) of a connected graph G of order n is the average of the distances between all pairs of vertices of G, i.e., μ(G) = ()⁻¹ Σ_{x,y}⊂V(G) d_G(x, y), where V(G) denotes the vertex set of G and d_G(x, y) is the distance between x and y. We prove that every connected graph of order n and minimum degree δ has a spanning tree T with average distance at most . We give improved bounds for K₃‐free graphs, C₄‐free graphs, and for graphs of given girth. © 2000 John Wiley & Sons, Inc. J Graph Theory 33: 1–13, 2000     

On long cycles in a 2-connected bipartite graph

Letk be an integer withk 2. LetG = (A, B; E) be a 2-connected bipartite graph. Supposed(x) + d(y) k + 1 for every pair of non-adjacent verticesx andy. ThenG contains a cycle of length at leastmin(2a, 2k) wherea = min(|A|,|B|), unlessG is one of some known exceptions. We conjecture that if|A| = |B| andd(x) + d(y) k + 1 for every pair of non-adjacent verticesx andy withx A andy B, thenG contains a cycle of length at leastmin(2a, 2k).     

Embedding graphs with bounded degree in sparse pseudorandom graphs

In this paper, we show the equivalence of somequasi-random properties for sparse graphs, that is, graphsG with edge densityp=|E(G)|/( ₂ ⁿ )=o(1), whereo(1)→0 asn=|V(G)|→∞. Our main result (Theorem 16) is the following embedding result. For a graphJ, writeN _J(x) for the neighborhood of the vertexx inJ, and letδ(J) andΔ(J) be the minimum and the maximum degree inJ. LetH be atriangle-free graph and setd _H=max{δ(J):J⊆H}. Moreover, putD _H=min{2d _H,Δ(H)}. LetC>1 be a fixed constant and supposep=p(n)≫n ⁻¹ D _H. We show that ifG is such that

Panconnected graphs II

A graphG of orderp is said to bepanconnected if for each pairu, v of vertices ofG, there exists a,u, v-path of lengthl inG, for eachl such that d_G(u, v)lp – 1, whered _G (u, v) denotes the length of a shortestu, v-path inG. Three conditions are shown to be sufficient for a graphG of orderp to be panconnected: (1) the degree of each vertex ofG is at least (p+2)/2; (2) the sum of the degrees of each pair of nonadjacent vertices ofG is at least (3p–2)/2; (3) the graphG has at least edges. It is also shown that each of these conditions is best possible. Additional results on panconnectedness are obtained including a characterization of those completen-partite graphs which are panconnected.     

Maximum hitting time for random walks on graphs

For x and y vertices of a connected graph G, let T_G(x, y) denote the expected time before a random walk starting from x reaches y. We determine, for each n > 0, the n-vertex graph G and vertices x and y for which T_G(x, y) is maximized. the extremal graph consists of a clique on ?(2n + 1)/3?) (or ?)(2_n ? 2)/3?) vertices, including x, to which a path on the remaining vertices, ending in y, has been attached; the expected time T_G(x, y) to reach y from x in this graph is approximately 4_n³/27.     

The group PSL(2,q) is not the multiplication group of a loop

ABSTRACT

We introduce the notion of weak transitivity for torsion-free abelian groups. A torsion-free abelian group G is called weakly transitive if for any pair of elements x, y ∈ G and endomorphisms ?, ψ ∈ End(G) such that x? = y, yψ = x, there exists an automorphism of G mapping x onto y. It is shown that every suitable ring can be realized as the endomorphism ring of a weakly transitive torsion-free abelian group, and we characterize up to a number-theoretical property the separable weakly transitive torsion-free abelian groups.     

The connectivities of adjacent tree graphs

The connectivity and the circuit rank of a graphG are denoted byx(G) and, respectively. It is shown that ifH is the adjacent tree graph of a simple connected graphG, thenx(H)=2.     

Curves in Grothendieck Categories

Noncommutative projective geometry studies noncommutative graded rings by replacing the variety by a suitable Grothendieck category. One way of studying the resulting category is to examine the full subcategories which behave like curves on a commutative variety. Smith and Zhang initiated such a study by considering the subcategory generated by a particular type of module they called a pure curve module in good position. This paper generalizes their construction by allowing more general modules. The resulting category is shown to be categorically equivalent to a quotient of the category of graded modules over a graded ring. In the course of defining the category equivalence, several dimensions, including projective, injective and Krull dimensions, are calculated. In particular, this extension allows examination of the category created from a line module over more general AS-regular rings than those considered by Smith and Zhang. For instance, suppose that C is a generic line module over R _d , Stafford's Sklyanin-like algebra. Let C denote the category C generates. Then C is equivalent to the category of graded k[x, y]/(x ² – y ²) modules under the Z × Z/2Z-grading where deg(x) = (–1, 0) and deg(y) = (–1,1).     

On a conjecture of Siegel

Letk be an algebraically closed field of characteristic 0 and letf(x, y)k[t][x,y] be a polynomial in two variables with coefficients ink[t]. One is interested in solving the equationf(x,y)=0 with polynomialsx,yk[t]. Two solutions(x,y), (x, y) areproportional ifx/x andy/y are non-zero constants ink and a solution(x,y) isprimitive if the polynomialsx andy have no common root. The main result of this paper is that for a certain class of polynomialsf, which includes Thue equations with sufficiently lacunary exponents, the number of non-proportional, primitive solutions is bounded solely in terms of the number of monomials appearing in the polynomialf(x,y). This verifies the analogue of a conjecture of Siegel for this class of polynomials. The proof is an application of theabc-theorem in function fields to certain determinantal varieties arising from the elimination of the coefficients of the polynomialf(x,y), together with an inductive argument on the numberr of monomials inf(x,y).