A note on transformations of edge colorings of bipartite graphs

The author and A. Mirumian proved the following theorem: Let G be a bipartite graph with maximum degree Δ and let t,n be integers, tnΔ. Then it is possible to obtain, from one proper edge t-coloring of G, any proper edge n-coloring of G using only transformations of 2-colored and 3-colored subgraphs such that the intermediate colorings are also proper. In this note we show that if t>Δ then we can transform f to g using only transformations of 2-colored subgraphs. We also correct the algorithm suggested in [A.S. Asratian, Short solution of Kotzig's problem for bipartite graphs, J. Combin. Theory Ser. B 74 (1998) 160–168] for transformation of f to g in the case when t=n=Δ and G is regular.     

disjoint cycles containing specified independent vertices

Let k1 be an integer and G be a graph of order n3k satisfying the condition that σ₂(G)n+k-1. Let v₁,…,v_k be k independent vertices of G, and suppose that G has k vertex-disjoint triangles C₁,…,C_k with v_iV(C_i) for all 1ik.Then G has k vertex-disjoint cycles such that

(i) for all 1ik.

(ii) , and

(iii) At least k-1 of the k cycles are triangles.

The condition of degree sum σ₂(G)n+k-1 is sharp.

On the number of independent chorded cycles in a graph

Hajnal and Corrádi proved that any simple graph on at least 3k vertices with minimal degree at least 2k contains k independent cycles. We prove the analogous result for chorded cycles. Let G be a simple graph with |V(G)|4k and minimal degree δ(G)3k. Then G contains k independent chorded cycles. This result is sharp.     

Minimum degree conditions for -linked graphs

For a fixed multigraph H with vertices w₁,…,w_m, a graph G is H-linked if for every choice of vertices v₁,…,v_m in G, there exists a subdivision of H in G such that v_i is the branch vertex representing w_i (for all i). This generalizes the notions of k-linked, k-connected, and k-ordered graphs.Given a connected multigraph H with k edges and minimum degree at least two and n7.5k, we determine the least integer d such that every n-vertex simple graph with minimum degree at least d is H-linked. This value D(H,n) appears to equal the least integer d^′ such that every n-vertex graph with minimum degree at least d^′ is b(H)-connected, where b(H) is the maximum number of edges in a bipartite subgraph of H.     

Orthogonal graphs of odd characteristic and their automorphisms

The (isotropic) orthogonal graph O(2ν+δ,q) over of odd characteristic, where ν1 and δ=0,1 or 2 is introduced. When ν=1, O(21+δ,q) is a complete graph. When ν2, O(2ν+δ,q) is strongly regular and its parameters are computed, as well as its chromatic number. The automorphism groups of orthogonal graphs are also determined.     

Steiner centers and Steiner medians of graphs

Let G be a connected graph and S a set of vertices of G. The Steiner distance of S is the smallest number of edges in a connected subgraph of G that contains S and is denoted by d_G(S) or d(S). The Steiner n-eccentricity e_n(v) and Steiner n-distance d_n(v) of a vertex v in G are defined as e_n(v)=max{d(S)| SV(G), |S|=n and vS} and d_n(v)=∑{d(S)| SV(G), |S|=n and vS}, respectively. The Steiner n-center C_n(G) of G is the subgraph induced by the vertices of minimum n-eccentricity. The Steiner n-median M_n(G) of G is the subgraph induced by those vertices with minimum Steiner n-distance. Let T be a tree. Oellermann and Tian [O.R. Oellermann, S. Tian, Steiner centers in graphs, J. Graph Theory 14 (1990) 585–597] showed that C_n(T) is contained in C_n+1(T) for all n2. Beineke et al. [L.W. Beineke, O.R. Oellermann, R.E. Pippert, On the Steiner median of a tree, Discrete Appl. Math. 68 (1996) 249–258] showed that M_n(T) is contained in M_n+1(T) for all n2. Then, Oellermann [O.R. Oellermann, On Steiner centers and Steiner medians of graphs, Networks 34 (1999) 258–263] asked whether these containment relationships hold for general graphs. In this note we show that for every n2 there is an infinite family of block graphs G for which C_n(G)C_n+1(G). We also show that for each n2 there is a distance–hereditary graph G such that M_n(G)M_n+1(G). Despite these negative examples, we prove that if G is a block graph then M_n(G) is contained in M_n+1(G) for all n2. Further, a linear time algorithm for finding the Steiner n-median of a block graph is presented and an efficient algorithm for finding the Steiner n-distances of all vertices in a block graph is described.     

Spanning 3-colourable subgraphs of small bandwidth in dense graphs

A conjecture by Bollobás and Komlós states the following: For every γ>0 and integers r2 and Δ, there exists β>0 with the following property. If G is a sufficiently large graph with n vertices and minimum degree at least and H is an r-chromatic graph with n vertices, bandwidth at most βn and maximum degree at most Δ, then G contains a copy of H.This conjecture generalises several results concerning sufficient degree conditions for the containment of spanning subgraphs. We prove the conjecture for the case r=3.     

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}.     

Asymptotic behavior of solutions of semilinear elliptic equations near an isolated singularity

We consider the semilinear elliptic equation Δu=h(u) in Ω{0}, where Ω is an open subset of (N2) containing the origin and h is locally Lipschitz continuous on [0,∞), positive in (0,∞). We give a complete classification of isolated singularities of positive solutions when h varies regularly at infinity of index q(1,C_N) (that is, lim_u→∞h(λu)/h(u)=λ^q, for every λ>0), where C_N denotes either N/(N−2) if N3 or ∞ if N=2. Our result extends a well-known theorem of Véron for the case h(u)=u^q.     

The perfect matching polytope and solid bricks

The perfect matching polytope of a graph G is the convex hull of the set of incidence vectors of perfect matchings of G. Edmonds (J. Res. Nat. Bur. Standards Sect. B 69B 1965 125) showed that a vector x in Q^E belongs to the perfect matching polytope of G if and only if it satisfies the inequalities: (i) x0 (non-negativity), (ii) x(∂(v))=1, for all vV (degree constraints) and (iii) x(∂(S))1, for all odd subsets S of V (odd set constraints). In this paper, we characterize graphs whose perfect matching polytopes are determined by non-negativity and the degree constraints. We also present a proof of a recent theorem of Reed and Wakabayashi.     

On shredders and vertex connectivity augmentation

We consider the following problem: given a k-(node) connected graph G find a smallest set F of new edges so that the graph G+F is (k+1)-connected. The complexity status of this problem is an open question. The problem admits a 2-approximation algorithm. Another algorithm due to Jordán computes an augmenting edge set with at most (k−1)/2 edges over the optimum. CV(G) is a k-separator (k-shredder) of G if |C|=k and the number b(C) of connected components of G−C is at least two (at least three). We will show that the problem is polynomially solvable for graphs that have a k-separator C with b(C)k+1. This leads to a new splitting-off theorem for node connectivity. We also prove that in a k-connected graph G on n nodes the number of k-shredders with at least p components (p3) is less than 2n/(2p−3), and that this bound is asymptotically tight.     

A supplement to the Davis–Gut law

Let {X,X_n;n1} be a sequence of i.i.d. real-valued random variables and set , n1. Let h() be a positive nondecreasing function such that . Define Lt=log_emax{e,t} for t0. In this note we prove that

if and only if E(X)=0 and E(X²)=1, where , t1. When h(t)≡1, this result yields what is called the Davis–Gut law. Specializing our result to h(t)=(Lt)^r, 0<r1, we obtain an analog of the Davis–Gut law.     

Generating random correlation matrices based on partial correlations

A d-dimensional positive definite correlation matrix R=(ρ_ij) can be parametrized in terms of the correlations ρ_i,i+1 for i=1,…,d-1, and the partial correlations ρ_{ij|i+1,…j-1} for j-i2. These parameters can independently take values in the interval (-1,1). Hence we can generate a random positive definite correlation matrix by choosing independent distributions F_ij, 1i<jd, for these parameters. We obtain conditions on the F_ij so that the joint density of (ρ_ij) is proportional to a power of det(R) and hence independent of the order of indices defining the sequence of partial correlations. As a special case, we have a simple construction for generating R that is uniform over the space of positive definite correlation matrices. As a byproduct, we determine the volume of the set of correlation matrices in -dimensional space. To prove our results, we obtain a simple remarkable identity which expresses det(R) as a function of ρ_i,i+1 for i=1,…,d-1, and ρ_{ij|i+1,…j-1} for j-i2.     

On the list dynamic coloring of graphs

A proper vertex coloring of a graph G is called a dynamic coloring if for every vertex v of degree at least 2, the neighbors of v receive at least two different colors. Assume that is the minimum number k such that for every list assignment of size k to each vertex of G, there is a dynamic coloring of G such that every vertex is colored with a color from its list. In this paper, it is proved that if G is a graph with no component isomorphic to C₅ and Δ(G)≥3, then , where Δ(G) is the maximum degree of G. This generalizes a result due to Lai, Montgomery and Poon which says that under the same assumptions χ₂(G)≤Δ(G)+1. Among other results, we determine , for every natural number n.     

Arithmetic progressions consisting of unlike powers

In this paper we present some new results about unlike powers in arithmetic progression. We prove among other things that for given k 4 and L 3 there are only finitely many arithmetic progressions of the form with x_i , gcd(x₀, x_l) = 1 and 2 l_i L for i = 0, 1, …, k − 1. Furthermore, we show that, for L = 3, the progression (1, 1,…, 1) is the only such progression up to sign. Our proofs involve some well-known theorems of Faltings [9], Darmon and Granville [6] as well as Chabauty's method applied to superelliptic curves.     

Uniform Kazhdan constant for some families of linear groups

Let R be a ring generated by l elements with stable range r. Assume that the group EL_d(R) has Kazhdan constant ₀>0 for some dr+1. We prove that there exist (₀,l)>0 and , s.t. for every nd, EL_n(R) has a generating set of order k and a Kazhdan constant larger than . As a consequence, we obtain for where n3, a Kazhdan constant which is independent of n w.r.t. generating set of a fixed size.     

On the code generated by the incidence matrix of points and k-spaces in and its dual

In this paper, we study the p-ary linear code C_k(n,q), q=p^h, p prime, h1, generated by the incidence matrix of points and k-dimensional spaces in PG(n,q). For kn/2, we link codewords of C_k(n,q)C_k(n,q) of weight smaller than 2q^k to k-blocking sets. We first prove that such a k-blocking set is uniquely reducible to a minimal k-blocking set, and exclude all codewords arising from small linear k-blocking sets. For k<n/2, we present counterexamples to lemmas valid for kn/2. Next, we study the dual code of C_k(n,q) and present a lower bound on the weight of the codewords, hence extending the results of Sachar [H. Sachar, The F_p span of the incidence matrix of a finite projective plane, Geom. Dedicata 8 (1979) 407–415] to general dimension.     

A new kind of graph coloring

A proper k-coloring C¹,C²,…,C^k of a graph G is called strong if, for every vertex uV(G), there exists an index i{1,2,…,k} such that u is adjacent to every vertex of Cⁱ. We consider classes of strongly k-colorable graphs and show that the recognition problem of is NP-complete for every k4, but it is polynomial-time solvable for k=3. We give a characterization of in terms of forbidden induced subgraphs. Finally, we solve the problem of uniqueness of a strong 3-coloring.     

On value sets of polynomials over a field

Let F be any field. Let p(F) be the characteristic of F if F is not of characteristic zero, and let p(F)=+∞ otherwise. Let A₁,…,A_n be finite nonempty subsets of F, and let

with k{1,2,3,…}, a₁,…,a_nF{0} and degg<k. We show that

When kn and |A_i|i for i=1,…,n, we also have

consequently, if nk then for any finite subset A of F we have

In the case n>k, we propose a further conjecture which extends the Erdős–Heilbronn conjecture in a new direction.