The minimum diameter of orientations of complete multipartite graphs

Given a graphG, letB be the family of strong orientations ofG, and define A pair {p,q} of integers is called aco-pair if 1 p q . A multiset {p, q, r} of positive integers is called aco-triple if {p, q} and {p, r} are co-pairs. LetK(p₁, p₂,..., p_n) denote the completen-partite graph havingp _i vertices in theith partite set.In this paper, we show that if {p ₁, p₂,...,p_n} can be partitioned into co-pairs whenn is even, and into co-pairs and a co-triple whenn is odd, then(K(p₁, p₂,..., p_n)) = 2 provided that (n,p ₁, p₂, p₃, p₄) (4, 1, 1, 1, 1). This substantially extends a result of Gutin [3] and a result of Koh and Tan [4].     

A Sharp Threshold for k‐Colorability

Let k be a fixed integer and f_k(n, p) denote the probability that the random graph G(n, p) is k‐colorable. We show that for k≥3, there exists d_k(n) such that for any ϵ>0, (1) As a result we conclude that for sufficiently large n the chromatic number of G(n, d/n) is concentrated in one value for all but a small fraction of d>1. ©1999 John Wiley & Sons, Inc. Random Struct. Alg., 14, 63–70, 1999     

Moment conditions and support theorems for Radon transforms on affine Grassmann manifolds

Let G(p,n) and G(q,n) be the affine Grassmann manifolds of p- and q-planes in Rⁿ, respectively, and let be the Radon transform from smooth functions on G(p,n) to smooth functions on G(q,n) arising from the inclusion incidence relation. When p<q and dimG(p,n)=dimG(p,n), we present a range characterization theorem for via moment conditions. We then use this range result to prove a support theorem for . This complements a previous range characterization theorem for via differential equations when dimG(p,n)<dimG(p,n). We also present a support theorem in this latter case.     

Graphs of Large Girth with Prescribed Partial Circular Colourings

This paper completes the constructive proof of the following result: Suppose p/q2 is a rational number, A is a finite set and f₁,f₂,···,f_n are mappings from A to {0,1,···,p–1}. Then for any integer g, there is a graph G=(V,E) of girth at least g with such that G has exactly n (p,q)-colourings (up to equivalence) g₁,g₂,···,g_n, and each g_i is an extension of f_i. A probabilistic proof of this result was given in [8]. A constructive proof of the case p/q3 was given in [7].This research was partially supported by the National Science Council under grant NSC91-2115-M-110-004     

An application of the regularity lemma in generalized Ramsey theory

Given graphs G and H, an edge coloring of G is called an (H,q)‐coloring if the edges of every copy of H ? G together receive at least q colors. Let r(G,H,q) denote the minimum number of colors in a (H,q)‐coloring of G. In 9 Erd?s and Gyárfás studied r(K_n,K_p,q) if p and q are fixed and n tends to infinity. They determined for every fixed p the smallest q (denoted by q_lin) for which r(K_n,K_p,q) is linear in n and the smallest q (denoted by q_quad) for which r(K_n,K_p,q) is quadratic in n. They raised the problem of determining the smallest q for which we have . In this paper by using the Regularity Lemma we show that if , then we have . © 2003 Wiley Periodicals, Inc. J Graph Theory 44: 39–49, 2003     

The Diameter of Sparse Random Graphs

We consider the diameter of a random graph G(n, p) for various ranges of p close to the phase transition point for connectivity. For a disconnected graph G, we use the convention that the diameter of G is the maximum diameter of its connected components. We show that almost surely the diameter of random graph G(n, p) is close to if np → ∞. Moreover if , then the diameter of G(n, p) is concentrated on two values. In general, if , the diameter is concentrated on at most 21/c₀ + 4 values. We also proved that the diameter of G(n, p) is almost surely equal to the diameter of its giant component if np > 3.6.     

A division problem for\bar \partial _b - CLOSED forms

LetG ₁,…,G_m be bounded holomorphic functions in a strictly pseudoconvex domainD such that . We prove that for each (0,q)-form ϕ inL ^p(∂D), 1<p<∞, there are formsu ₁, …,u _m inL ^p(∂D) such that ΣG _ju_j=ϕ. This generalizes previous results forq=0. The proof consists in delicate estimates of integral representation formulas of solutions and relies on a certainT1 theorem due to Christ and Journé. For (0,n−1)-forms there is a simpler proof that also gives the result forp=∞. Restricted to one variable this is precisely the corona theorem. The author was partially supported by the Swedish Natural Research Council.     

Random graphs and covering graphs of posets

For a graph G, we define c(G) to be the minimal number of edges we must delete in order to make G into a covering graph of some poset. We prove that, if p=n ^-1+(n) ,where (n) is bounded away from 0, then there is a constant k ₀>0 such that, for a.e. G _p , c(G _p )k ₀ n ¹⁺⁽ⁿ⁾ .In other words, to make G _p into a covering graph, we must almost surely delete a positive constant proportion of the edges. On the other hand, if p=n ^-1+(n) , where (n)0, thenc(G _p )=o(n ¹⁺⁽ⁿ⁾ ), almost surely.Partially supported by MCS Grant 8104854.     

Graham’s pebbling conjecture on product of thorn graphs of complete graphs

The pebbling number of a graph G, f(G), is the least n such that, no matter how n pebbles are placed on the vertices of G, we can move a pebble to any vertex by a sequence of pebbling moves, each move taking two pebbles off one vertex and placing one on an adjacent vertex. Let p₁,p₂,…,p_n be positive integers and G be such a graph, V(G)=n. The thorn graph of the graph G, with parameters p₁,p₂,…,p_n, is obtained by attaching p_i new vertices of degree 1 to the vertex u_i of the graph G, i=1,2,…,n. Graham conjectured that for any connected graphs G and H, f(G×H)≤f(G)f(H). We show that Graham’s conjecture holds true for a thorn graph of the complete graph with every by a graph with the two-pebbling property. As a corollary, Graham’s conjecture holds when G and H are the thorn graphs of the complete graphs with every .     

Transformation graph

The transformation graph G^-+- of a graph G is the graph with vertex set V(G)E(G), in which two vertices u and v are joined by an edge if one of the following conditions holds: (i) u,vV(G) and they are not adjacent in G, (ii) u,vE(G) and they are adjacent in G, (iii) one of u and v is in V(G) while the other is in E(G), and they are not incident in G. In this paper, for any graph G, we determine the connectivity and the independence number of G^-+-. Furthermore, for a graph G of order n4, we show that G^-+- is hamiltonian if and only if G is not isomorphic to any graph in {2K₁+K₂,K₁+K₃}{K_1,n-1,K_1,n-1+e,K_1,n-2+K₁}.     

The energy of unitary cayley graphs

A graph G of order n is called hyperenergetic if E(G)>2n-2, where E(G) denotes the energy of G. The unitary Cayley graph X_n has vertex set Z_n={0,1,2,…,n-1} and vertices a and b are adjacent, if gcd(a-b,n)=1. These graphs have integral spectrum and play an important role in modeling quantum spin networks supporting the perfect state transfer. We show that the unitary Cayley graph X_n is hyperenergetic if and only if n has at least two prime factors greater than 2 or at least three distinct prime factors. In addition, we calculate the energy of the complement of unitary Cayley graph and prove that is hyperenergetic if and only if n has at least two distinct prime factors and n≠2p, where p is a prime number. By extending this approach, for every fixed , we construct families of k hyperenergetic non-cospectral integral circulant n-vertex graphs with equal energy.     

Modularity of Erdős‐Rényi random graphs

For a given graph G, each partition of the vertices has a modularity score, with higher values indicating that the partition better captures community structure in G. The modularity q^?(G) of the graph G is defined to be the maximum over all vertex partitions of the modularity score, and satisfies 0 ≤ q^?(G)<1. Modularity is at the heart of the most popular algorithms for community detection. We investigate the behaviour of the modularity of the Erd?s‐Rényi random graph G_n,p with n vertices and edge‐probability p. Two key findings are that the modularity is 1+o(1) with high probability (whp) for np up to 1+o(1) and no further; and when np ≥ 1 and p is bounded below 1, it has order (np)^?1/2 whp, in accord with a conjecture by Reichardt and Bornholdt in 2006. We also show that the modularity of a graph is robust to changes in a few edges, in contrast to the sensitivity of optimal vertex partitions.     

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

Removable singularities of weak solutions to linear partial differential equations

Suppose that P(x, D) is a linear differential operator of order m > 0 with smooth coefficients whose derivatives up to order m are continuous functions in the domain G ⁿ (n 1), 1 < p > , s > 0, and q=p/(p – 1). In this paper, we show that if n, m, p, and s satisfy the two-sided bound 0 n – q(m – s)< n, then for a weak solution of the equation P(x, D)u=0 from the Sharpley-DeVore class C _p ^s (G)_loc, any closed set in G is removable if its Hausdorff measure of order n – q(m – s) is finite. This result strengthens the well-known result of Harvey and Polking on removable singularities of weak solutions to the equation P(x, D)_u=0 from the Sobolev classes and extends it to the case of noninteger orders of smoothness.Translated from Matematicheskie Zametki, vol. 77, no. 4, 2005, pp. 584–591.Original Russian Text Copyright © 2005 by A. V. Pokrovskii.This revised version was published online in April 2005 with a corrected issue number.     

Perfect matchings in random uniform hypergraphs

Let ??_k(n, p) be the random k‐uniform hypergraph on V = [n] with edge probability p. Motivated by a theorem of Erd?s and Rényi 7 regarding when a random graph G(n, p) = ??₂(n, p) has a perfect matching, the following conjecture may be raised. (See J. Schmidt and E. Shamir 16 for a weaker version.) Conjecture. Let k|n for fixed k ≥ 3, and the expected degree d(n, p) = p(). Then (Erd?s and Rényi 7 proved this for G(n, p).) Assuming d(n, p)/n^1/2 → ∞, Schmidt and Shamir 16 were able to prove that ??_k(n, p) contains a perfect matching with probability 1 ? o(1). Frieze and Janson 8 showed that a weaker condition d(n, p)/n^1/3 → ∞ was enough. In this paper, we further weaken the condition to A condition for a similar problem about a perfect triangle packing of G(n, p) is also obtained. A perfect triangle packing of a graph is a collection of vertex disjoint triangles whose union is the entire vertex set. Improving a condition p ≥ cn^?2/3+1/15 of Krivelevich 12 , it is shown that if 3|n and p ? n^?2/3+1/18, then © 2003 Wiley Periodicals, Inc. Random Struct. Alg., 23: 111–132, 2003     

A Sharp Upper Bound for the Number of Spanning Trees of a Graph

Let G = (V,E) be a simple graph with n vertices, e edges and d₁ be the highest degree. Further let λ_i, i = 1,2,...,n be the non-increasing eigenvalues of the Laplacian matrix of the graph G. In this paper, we obtain the following result: For connected graph G, λ₂ = λ₃ = ... =  λ_n-1 if and only if G is a complete graph or a star graph or a (d₁,d₁) complete bipartite graph. Also we establish the following upper bound for the number of spanning trees of G on n, e and d₁ only:

Pan-connectedness of graphs with large neighborhood unions

Let G be a simple graph with n vertices. For any , let , and , and and u and v has distance 2 in E(G)}. Let l ≥ 1 be an integer. A graph G on n ≥ l vertices is [l, n]-pan-connected if for any , and any integer m with l ≤ m ≤ n, G has a (u, v)-path of length m. In 1998, Wei and Zhu (Graphs Combinatorics 14:263–274, 1998) proved that for a three-connected graph on n ≥ 7 vertices, if NC(G) ≥ n − δ(G) + 1, then G is [6, n]-pan-connected. They conjectured that such graphs should be [5, n]-pan-connected. In this paper, we prove that for a three-connected graph on n ≥ 7 vertices, if NC ₂(G) ≥ n − δ(G) + 1, then G is [5, n]-pan-connected. Consequently, the conjecture of Wei and Zhu is proved as NC ₂(G) ≥ NC(G). Furthermore, we show that the lower bound is best possible and characterize all 2-connected graphs with NC ₂(G) ≥ n − δ(G) + 1 which are not [4, n]-pan-connected.      

The phase transition in random graphs: A simple proof

The classical result of Erd?s and Rényi asserts that the random graph G(n,p) experiences sharp phase transition around \begin{align*}p=\frac{1}{n}\end{align*} – for any ε > 0 and \begin{align*}p=\frac{1-\epsilon}{n}\end{align*}, all connected components of G(n,p) are typically of size O_ε(log n), while for \begin{align*}p=\frac{1+\epsilon}{n}\end{align*}, with high probability there exists a connected component of size linear in n. We provide a very simple proof of this fundamental result; in fact, we prove that in the supercritical regime \begin{align*}p=\frac{1+\epsilon}{n}\end{align*}, the random graph G(n,p) contains typically a path of linear length. We also discuss applications of our technique to other random graph models and to positional games. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2013     

On the Diameter of Wenger Graphs

Let q be a prime power, the field of q elements, and n≥1 a positive integer. The Wenger graph W _n (q) is defined as follows: the vertex set of W _n (q) is the union of two copies P and L of (n+1)-dimensional vector spaces over , with two vertices (p ₁,p ₂,…,p _n+1)∈P and [l ₁,l ₂,…,l _n+1]∈L being adjacent if and only if l _i +p _i =p ₁ l _i−1 for 2≤i≤n+1. Graphs W _n (q) have several interesting properties. In particular, it is known that when connected, their diameter is at most 2n+2. In this note we prove that the diameter of connected Wenger graphs is 2n+2 under the assumption that 1≤n≤q−1.     

On the multiplicity of the eigenvalues of a graph

Given a graph G with characteristic polynomial ϕ(t), we consider the ML-decomposition ϕ(t) = q ₁(t)q ₂(t)² ... q _m (t)^m, where each q _i (t) is an integral polynomial and the roots of ϕ(t) with multiplicity j are exactly the roots of q _j (t). We give an algorithm to construct the polynomials q _i (t) and describe some relations of their coefficients with other combinatorial invariants of G. In particular, we get new bounds for the energy E(G) = |λ_i| of G, where λ₁, λ₂, ..., λ_n are the eigenvalues of G (with multiplicity). Most of the results are proved for the more general situation of a Hermitian matrix whose characteristic polynomial has integral coefficients. This work was done during a visit of the second named author to UNAM.