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.     

Extending matchings in planar graphs V

A graph G on at least 2n + 2 vertices in n-extendable if every set of n independent edges extends to (i.e., is a subset of) a perfect matching in G. It is known that no planar graph is 3-extendable. In the present paper we continue to study 2-extendability in the plane. Suppose independent edges e₁ and e₂ are such that the removal of their endvertices leaves at least one odd component C_o. The subgraph G[V(C_o) V(e₁) V(e₂)] is called a generalized butterfly (or gbutterfly). Clearly, a 2-extendable graph can contain no gbutterfly. The converse, however, is false.

We improve upon a previous result by proving that if G is 4-connected, locally connected and planar with an even number of vertices and has no gbutterfly, it is 2-extendable. Sharpness with respect to the various hypotheses of this result is discussed.     

Largest signless Laplacian spectral radius of uniform supertrees with diameter and pendent edges (vertices)

Let \mathbb{S}(m; d; k) be the set of k-uniform supertrees with m edges and diameter d; and S₁(m; d; k) be the k-uniform supertree obtained from a loose path u₁; e₁; u₂; e₂,..., u_d; e_d; u_d+1 with length d by attaching m — d edges at vertex u_{\lfloor d/2\rfloor +1}: In this paper, we mainly determine S₁(m; d; k) with the largest signless Laplacian spectral radius in \mathbb{S}(m; d; k) for 3≤d≤m –1: We also determine the supertree with the second largest signless Laplacian spectral radius in \mathbb{S}(m; 3; k): Furthermore, we determine the unique k-uniform supertree with the largest signless Laplacian spectral radius among all k-uniform supertrees with n vertices and pendent edges (vertices).     

Ramsey linear families and generalized subdivided graphs

The following results are obtained. (i) Let p, d, and k be fixed positive integers, and let G be a graph whose vertex set can be partitioned into parts V₁, V₂,…, V_a such that for each i at most d vertices in V₁ … V_i have neighbors in V_i+1 and r(K_k, V_i) p | V(G) |, where V_i denotes the subgraph of G induced by V_i. Then there exists a number c depending only on p, d, and k such that r(K_k, G)c | V(G) |. (ii) Let d be a positive integer and let G be a graph in which there is an independent set I V(G) such that each component of G−I has at most d vertices and at most two neighbors in I. Then r(G,G)c | V(G) |, where c is a number depending only on d. As a special case, r(G, G) 6 | V(G) | for a graph G in which all vertices of degree at least three are independent. The constant 6 cannot be replaced by one less than 4.     

Zagreb indices of graphs

The first Zagreb index M₁(G) is equal to the sum of squares of the degrees of the vertices, and the second Zagreb index M₂(G) is equal to the sum of the products of the degrees of pairs of adjacent vertices of the underlying molecular graph G. In this paper, we obtain lower and upper bounds on the first Zagreb index M₁(G) of G in terms of the number of vertices (n), number of edges (m), maximum vertex degree (Δ), and minimum vertex degree (δ). Using this result, we find lower and upper bounds on M₂(G). Also, we present lower and upper bounds on M₂(G) +M₂(G) in terms of n, m, Δ, and δ, where G denotes the complement of G. Moreover, we determine the bounds on first Zagreb coindex M₁(G) and second Zagreb coindex M₂(G). Finally, we give a relation between the first Zagreb index and the second Zagreb index of graph G.     

Some Ore-type Results for Matching and Perfect Matching in <Emphasis Type="Italic">k</Emphasis>-uniform Hypergraphs

Let S₁ and S₂ be two (k-1)-subsets in a k-uniform hypergraph H. We call S₁ and S₂ strongly or middle or weakly independent if H does not contain an edge e ∈ E(H) such that S₁ ∩ e ≠∅ and S₂ ∩ e ≠∅ or e ⊆ S₁ ∪ S₂ or e ⊇ S₁ ∪ S₂, respectively. In this paper, we obtain the following results concerning these three independence. (1) For any n ≥ 2k²-k and k ≥ 3, there exists an n-vertex k-uniform hypergraph, which has degree sum of any two strongly independent (k-1)-sets equal to 2n-4(k-1), contains no perfect matching; (2) Let d ≥ 1 be an integer and H be a k-uniform hypergraph of order n ≥ kd+(k-2)k. If the degree sum of any two middle independent (k-1)-subsets is larger than 2(d-1), then H contains a d-matching; (3) For all k ≥ 3 and sufficiently large n divisible by k, we completely determine the minimum degree sum of two weakly independent (k-1)-subsets that ensures a perfect matching in a k-uniform hypergraph H of order n.     

On the fourth power moment of Fourier coefficients of cusp form

Let a(n)be the Fourier coefficients of a holomorphic cusp form of weightκ=2n≥12 for the full modular group and A(x)=∑_(n≤x)a(n).In this paper,we establish an asymptotic formula of the fourth power moment of A(x)and prove that ∫T1A~4(x)dx=3/(64κπ~4)s_4;2()T~(2κ)+O(T~(2κ-δ_4+ε))with δ_4=1/8,which improves the previous result.     

On the Equation n_1n_2= n_3n_4 Restricted to Factor Closed Sets

We study the number of solutions N(B,F) of the diophantine equation n_1n_2 = n_3 n_4,where 1 ≤ n_1 ≤ B,1 ≤ n_3 ≤ B,n_2,n_4 ∈ F and F[1,B] is a factor closed set.We study more particularly the case when F={m = p_1~(ε1)···p_k~(εk),ε_j∈{0,1},1 ≤ j ≤ k},p_1,...,p_k being distinct prime numbers.     

An extended result of Kleitman and Saks concerning binary trees

In a recent paper, D.J. Kleitman and M.E. Saks gave a proof of Huang's conjecture on alphabetic binary trees.

Given a set E = {e_i}, I = 0, 1, 2, …, m and assigned positive weights to its elements and supposing the elements are indexed such that w(e₀) ≤ w(e₁) ≤ … ≤w (e_m), where w(e_i) is the weight of e_i, we call the following sequence E^* a ‘saw-tooth’ sequence

E^*=(e₀,e_m,e₁,…,e_j,e_m−j,…).

Huang's conjecture is: E^* is the most expensive sequence for alphabetic binary trees. This paper shows that this property is true for the L-restricted alphabetic binary trees, where L is the maximum length of the leaves and log₂(m + 1) ≤L≤m.     

Strong Laws of Large Numbers for Double Sums of Banach Space Valued Random Elements

For a double array {V_(m,n), m ≥ 1, n ≥ 1} of independent, mean 0 random elements in a real separable Rademacher type p(1 ≤ p ≤ 2) Banach space and an increasing double array {b_(m,n), m ≥1, n ≥ 1} of positive constants, the limit law ■ and in L_p as m∨n→∞ is shown to hold if ■ This strong law of large numbers provides a complete characterization of Rademacher type p Banach spaces. Results of this form are also established when 0 p ≤ 1 where no independence or mean 0 conditions are placed on the random elements and without any geometric conditions placed on the underlying Banach space.     

Constructions of (q, K, λ, t, Q) almost difference families

The concept of a (q, k, λ, t) almost difference family (ADF) has been introduced and studied by C. Ding and J. Yin as a useful generalization of the concept of an almost difference set. In this paper, we consider, more generally, (q, K, λ, t, Q)-ADFs, where K = {k₁, k₂, ..., k_r} is a set of positive integers and Q = (q₁, q₂,..., q_r) is a given block-size distribution sequence. A necessary condition for the existence of a (q, K, λ, t, Q)-ADF is given, and several infinite classes of (q, K, λ, t, Q)-ADFs are constructed.     

Equality of higher numerical ranges of matrices and a conjecture of Kippenhahn on Hermitian pencils

Let M_n be the algebra of all n × n complex matrices. For 1 k n, the kth numerical range of A M_n is defined by W_k(A) = (1/k)∑_j^k=1x_j^*Ax_j : x₁, …, x_k is an orthonormal set in ⁿ]. It is known that tr A/n = W_n(A) W_n−1(A) W₁(A). We study the condition on A under which W_m(A) = W_k(A) for some given 1 m < k n. It turns out that this study is closely related to a conjecture of Kippenhahn on Hermitian pencils. A new class of counterexamples to the conjecture is constructed, based on the theory of the numerical range.     

Measurements of edge-uncolorability

Cubic bridgeless graphs with chromatic index four are called uncolorable. We introduce parameters measuring the uncolorability of those graphs and relate them to each other. For k=2,3, let c_k be the maximum size of a k-colorable subgraph of a cubic graph G=(V,E). We consider r₃=|E|−c₃ and . We show that on one side r₃ and r₂ bound each other, but on the other side that the difference between them can be arbitrarily large. We also compare them to the oddness ω of G, the smallest possible number of odd circuits in a 2-factor of G. We construct cyclically 5-edge connected cubic graphs where r₃ and ω are arbitrarily far apart, and show that for each 1c<2 there is a cubic graph such that ωcr₃. For k=2,3, let ζ_k denote the largest fraction of edges that can be k-colored. We give best possible bounds for these parameters, and relate them to each other.     

On the minimal reducible bound for outerplanar and planar graphs

Let L be the set of all additive and hereditary properties of graphs. For P₁, P₂ L we define the reducible property R = P₁ P₂ as follows: G P₁P₂ if there is a bipartition (V₁, V₂) of V(G) such that V₁ P₁ and V₂ P₂. For a property P L, a reducible property R is called a minimal reducible bound for P if P R and for each reducible property R′, R′ R → P R′. It is proved that the class of all outerplanar graphs has exactly two minimal reducible bounds in L. Some related problems for planar graphs are discussed.     

All trees are 1-embeddable and all except stars are 2-embeddable

A graph G with n vertices is said to be embeddable (in its complement) if there is an automorphism φ of K_n such that E(G) ∩ E(φ(G))=. It is known that all trees T with n (≥2) vertices and T K_1,n−1 are embeddable. We say that G is 1-embeddable if, for every edge e, there is an automorphism φ of K_n such that E(G) ∩ E(φ(G))={e};and that it is 2-embeddable if,for every pair e₁, e₂ of edges, there is an automorphism φ of K_n such that E(G) ∩ E(φ(G))={e₁, e₂}. We prove here that all trees with n (3) vertices are 1-embeddable; and that all trees T with n (4) vertices and T K_1,n−1 are 2-embeddable. In a certain sense, this result is sharp.     

Homogenization of Elliptic Problems with Neumann Boundary Conditions in Non-smooth Domains

We consider a family of second-order elliptic operators {L_ε} in divergence form with rapidly oscillating and periodic coefficients in Lipschitz and convex domains in R~n. We are able to show that the uniform W~(1,p) estimate of second order elliptic systems holds for 2n/(n+1)-δ p 2n/(n-1)+ δ where δ 0 is independent of ε and the ranges are sharp for n = 2, 3. And for elliptic equations in Lipschitz domains, the W~(1,p) estimate is true for 3/2-δ p 3 + δ if n ≥ 4, similar estimate was extended to convex domains for 1 p ∞.     

ContributionsChromatic Ramsey numbers

Suppose G is a graph. The chromatic Ramsey number r_c(G) of G is the least integer m such that there exists a graph F of chromatic number m for which the following is true: for any 2-colouring of the edges of F there is a monochromatic subgraph isomorphic to G. Let M_n = min[r_c(G): χ(G) = n]. It was conjectured by Burr et al. (1976) that M_n = (n − 1)² + 1. This conjecture has been confirmed previously for n 4. In this paper, we shall prove that the conjecture is true for n = 5. We shall also improve the upper bounds for M₆ and M₇.     

Length-bounded disjoint paths in planar graphs

The following problem is considered: given: an undirected planar graph G=(V,E) embedded in , distinct pairs of vertices {r₁,s₁},…,{r_k,s_k} of G adjacent to the unbounded face, positive integers b₁,…,b_k and a function ; find: pairwise vertex-disjoint paths P₁,…,P_k such that for each i=1,…,k, P_i is a r_i−s_i-path and the sum of the l-length of all edges in P_i is at most b_i. It is shown that the problem is NP-hard in the strong sense. A pseudo-polynomial-time algorithm is given for the case of k=2.     

An infinite family of cubic edge- but not vertex-transitive graphs

An infinite family of cubic edge- but not vertex-transitive graphs is constructed. The graphs are obtained as regular -covers of K_3,3 where n=p₁^e₁p₂^e₂p_k^e_k where p_i are distinct primes congruent to 1 modulo 3, and e_i1. Moreover, it is proved that the Gray graph (of order 54) is the smallest cubic edge- but not vertex-transitive graph.     

带松弛条件的图的强边着色

给定两个非负整数s和t,图G的(s,t)-松弛强k边着色可表示为映射c：E(G)→[k],这个映射满足对G中的任意一条边e,颜色c(e)在e的1-邻域中最多出现s次并且在e的2-邻域中最多出现t次。图G的(s,t)-松弛强边着色指数,记作χ'_(s,t)(G),表示使得图G有(s,t)-松弛强k边着色的最小k值。在图G中,如果mad(G) < 3并且Δ≤4,那么χ'_(1,0)(G)≤3Δ。并证明如果G是平面图,最大度Δ≥4并且围长最少为7,那么χ'_(1,0)(G)≤3Δ-1。